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"" -1 319 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 320 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 321 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 27 "An introduction to matrices" }} {PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }} {PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 19 "What is a matrix? " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 259 6 "ma trix" }{TEXT -1 114 " is a rectangular array of real (or complex) numb ers. We shall work with matrices which have real number entries. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 8 "Examples " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "A 2 " }{TEXT 263 1 "x " }{TEXT -1 19 " 3 matrix with two " }{TEXT 259 4 "rows" }{TEXT -1 11 " and three " }{TEXT 259 7 "columns" }{TEXT -1 36 ", and having real n umber entries: " }{XPPEDIT 18 0 "matrix([[2, 300, -4/7], [sqrt(7), P i/6, exp(-2)]]);" "6#-%'matrixG6#7$7%\"\"#\"$+$,$*&\"\"%\"\"\"\"\"(!\" \"F/7%-%%sqrtG6#F.*&%#PiGF-\"\"'F/-%$expG6#,$F(F/" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "A 2 " } {TEXT 264 1 "x" }{TEXT -1 71 " 2 matrix with two rows and two columns, and having integer entries: " }{XPPEDIT 18 0 "matrix([[17, -9], [3, 46]]);" "6#-%'matrixG6#7$7$\"#<,$\"\"*!\"\"7$\"\"$\"#Y" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "A 3 \+ " }{TEXT 265 1 "x" }{TEXT -1 74 " 3 matrix with three rows and three c olumns, and having rational entries: " }{XPPEDIT 18 0 "matrix([[13/45, -22/9, 56/87], [23/6, 32, 7/8],[5, 1, -73]])" "6#-%'matrixG6#7%7%*&\" #8\"\"\"\"#X!\"\",$*&\"#AF*\"\"*F,F,*&\"#cF*\"#()F,7%*&\"#BF*\"\"'F,\" #K*&\"\"(F*\"\")F,7%\"\"&F*,$\"#tF," }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The last two matrices \+ are " }{TEXT 259 15 "square matrices" }{TEXT -1 44 ", with the same nu mber of rows and columns. " }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "matrix([[x[1], x[2], x[3], x[4], x[5]]]);" "6#-%'matrix G6#7#7'&%\"xG6#\"\"\"&F)6#\"\"#&F)6#\"\"$&F)6#\"\"%&F)6#\"\"&" }{TEXT -1 9 " is a 1 " }{TEXT 266 1 "x" }{TEXT -1 14 " 5 matrix, or " } {TEXT 259 10 "row vector" }{TEXT -1 9 ", while " }{XPPEDIT 18 0 "matr ix([[x[1]], [x[2]], [x[3]], [x[4]]]);" "6#-%'matrixG6#7&7#&%\"xG6#\"\" \"7#&F)6#\"\"#7#&F)6#\"\"$7#&F)6#\"\"%" }{TEXT -1 9 " is a 4 " } {TEXT 267 1 "x" }{TEXT -1 14 " 1 matrix, or " }{TEXT 259 13 "column ve ctor" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "A general " }{TEXT 271 1 "m" }{TEXT -1 1 " " }{TEXT 268 1 "x" }{TEXT -1 1 " " }{TEXT 272 1 "n" }{TEXT -1 14 " matrix, with " }{TEXT 273 1 "m" }{TEXT -1 10 " rows and " }{TEXT 274 1 "n" }{TEXT -1 24 " columns, has the form: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "A = matrix([[a[1,1], a[1,2], a[1,3], ` . . . `, a[1,j], ` . . . `, a[1,n]], [a[2,1], a[2,2], a[2,3], ` . . . `, a[2,j], ` . . . `, a[2,n]], [a[3,1], a[3,2], a[3,3], ` . . . `, a[3,j], ` . . . `, \+ a[3,n]], [``, ``, ``, ``, ``, ``, ``], [a[i,1], a[i,2], a[i,3], ` . . \+ . `, a[i,j], ` . . . `, a[i,n]], [``, ``, ``, ``, ``, ``, ``], [a[m,1] , a[m,2], a[m,3], ` . . . `, a[m,j], ` . . . `, a[m,n]]]);" "6#/%\"AG- %'matrixG6#7)7)&%\"aG6$\"\"\"F-&F+6$F-\"\"#&F+6$F-\"\"$%(~.~.~.~G&F+6$ F-%\"jGF4&F+6$F-%\"nG7)&F+6$F0F-&F+6$F0F0&F+6$F0F3F4&F+6$F0F7F4&F+6$F0 F:7)&F+6$F3F-&F+6$F3F0&F+6$F3F3F4&F+6$F3F7F4&F+6$F3F:7)%!GFRFRFRFRFRFR 7)&F+6$%\"iGF-&F+6$FVF0&F+6$FVF3F4&F+6$FVF7F4&F+6$FVF:7)FRFRFRFRFRFRFR 7)&F+6$%\"mGF-&F+6$F]oF0&F+6$F]oF3F4&F+6$F]oF7F4&F+6$F]oF:" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "The first index " }{TEXT 269 1 "i" }{TEXT -1 36 " of each entry of th e matrix is the " }{TEXT 259 9 "row index" }{TEXT -1 23 ", and the sec ond index " }{TEXT 270 1 "j" }{TEXT -1 8 " is the " }{TEXT 259 12 "col umn index" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "We may also write: " }{XPPEDIT 18 0 "A=[a[i,j]][m,n]" "6#/%\"AG&7#&%\"aG6$%\"iG %\"jG6$%\"mG%\"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 87 "Addition and subtraction of matrices, and multiplicatio n of a matrix by a real number " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 95 "Matrices of the same \"size \", that is, with exactly the same number of rows and columns, can be \+ " }{TEXT 259 5 "added" }{TEXT -1 5 " and " }{TEXT 259 10 "subtracted" }{TEXT -1 53 " by adding or subtracting the corresponding entries. " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1,2],[3,4], [5,6]]) + matrix([[-1,3],[7,2],[5,-4]]) = matrix([[0,5],[10,6],[10,2 ]])" "6#/,&-%'matrixG6#7%7$\"\"\"\"\"#7$\"\"$\"\"%7$\"\"&\"\"'F*-F&6#7 %7$,$F*!\"\"F-7$\"\"(F+7$F0,$F.F7F*-F&6#7%7$\"\"!F07$\"#5F17$FBF+" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "matrix([[9, 2, 6], [5, 4, 5], [-1, 4, \+ 8]]) - matrix([[-5, 2, -3], [3, 3, -1], [1, -5, 2]])=matrix([[14, 0, 9 ], [2, 1, 6], [-2, 9, 6]])" "6#/,&-%'matrixG6#7%7%\"\"*\"\"#\"\"'7%\" \"&\"\"%F.7%,$\"\"\"!\"\"F/\"\")F2-F&6#7%7%,$F.F3F+,$\"\"$F37%F;F;,$F2 F37%F2,$F.F3F+F3-F&6#7%7%\"#9\"\"!F*7%F+F2F,7%,$F+F3F*F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "In general, if " }{XPPEDIT 18 0 "A=[a[i,j]][m,n]" "6#/%\"AG&7#&%\"aG6$% \"iG%\"jG6$%\"mG%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B=[b[i,j]][ m,n]" "6#/%\"BG&7#&%\"bG6$%\"iG%\"jG6$%\"mG%\"nG" }{TEXT -1 7 ", then \+ " }{XPPEDIT 18 0 "A+B = [a[i,j]+b[i,j]][m,n]" "6#/,&%\"AG\"\"\"%\"BGF& &7#,&&%\"aG6$%\"iG%\"jGF&&%\"bG6$F.F/F&6$%\"mG%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "A-B=[a[i,j]-b[i,j]][m,n]" "6#/,&%\"AG\"\"\"%\"BG!\" \"&7#,&&%\"aG6$%\"iG%\"jGF&&%\"bG6$F/F0F(6$%\"mG%\"nG" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "A matr ix " }{XPPEDIT 18 0 "A = [a[i,j]][m,n]" "6#/%\"AG&7#&%\"aG6$%\"iG%\"jG 6$%\"mG%\"nG" }{TEXT -1 8 " can be " }{TEXT 259 22 "multiplied by a sc alar" }{TEXT -1 18 " (or real number) " }{TEXT 275 1 "r" }{TEXT -1 21 " to give the matrix " }{XPPEDIT 18 0 "r*A = [r*a[i,j]][m,n]" "6#/*&% \"rG\"\"\"%\"AGF&&7#*&F%F&&%\"aG6$%\"iG%\"jGF&6$%\"mG%\"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 " For example, " }{XPPEDIT 18 0 "3*matrix([[-3,7],[1,4]])=matrix([[-9,21 ],[3,12]])" "6#/*&\"\"$\"\"\"-%'matrixG6#7$7$,$F%!\"\"\"\"(7$F&\"\"%F& -F(6#7$7$,$\"\"*F-\"#@7$F%\"#7" }{TEXT -1 7 ", and " }{XPPEDIT 18 0 " 1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([ [3,4],[2,-1],[0,8]])=matrix([[3/2,2],[1,-1/2],[0,4]])" "6#/-%'matrixG6 #7%7$\"\"$\"\"%7$\"\"#,$\"\"\"!\"\"7$\"\"!\"\")-F%6#7%7$*&F)F.F,F/F,7$ F.,$*&F.F.F,F/F/7$F1F*" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Matrix mul tiplication " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 116 "We start with the definition of the product of a row vector with a column vector having the same number of entries. \+ " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "matrix([[a[1], a [2], a[3], a[4]]])*matrix([[b[1]], [b[2]], [b[3]], [b[4]]]) = a[1]*b[1 ]+a[2]*b[2]+a[3]*b[3]+a[4]*b[4];" "6#/*&-%'matrixG6#7#7&&%\"aG6#\"\"\" &F+6#\"\"#&F+6#\"\"$&F+6#\"\"%F--F&6#7&7#&%\"bG6#F-7#&F<6#F07#&F<6#F37 #&F<6#F6F-,**&&F+6#F-F-&F<6#F-F-F-*&&F+6#F0F-&F<6#F0F-F-*&&F+6#F3F-&F< 6#F3F-F-*&&F+6#F6F-&F<6#F6F-F-" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 212 "Pair up the entries of t he first (row) vector with the entries of the second (column) vector. \+ The corresponding entries are multiplied together, and the resulting p roducts are added to give a single real number. " }}{PARA 0 "" 0 "" {TEXT -1 57 "This calculation is essentially the same as the standard \+ " }{TEXT 259 11 "dot product" }{TEXT -1 17 " of two vectors. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[4, 2, -1, 3]])*matrix([[3], [-5], [5], [7]]) = ``(4)*`.`*``(3)+``(2)*`.`*``(-5)+``(-1)*`.`*``(5)+``(3)*`.`*``(7);" " 6#/*&-%'matrixG6#7#7&\"\"%\"\"#,$\"\"\"!\"\"\"\"$F--F&6#7&7#F/7#,$\"\" &F.7#F67#\"\"(F-,**(-%!G6#F*F-%\".GF--F=6#F/F-F-*(-F=6#F+F-F?F--F=6#,$ F6F.F-F-*(-F=6#,$F-F.F-F?F--F=6#F6F-F-*(-F=6#F/F-F?F--F=6#F9F-F-" } {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 "= " }{XPPEDIT 18 0 "12-10-5+21=18" "6#/,*\"#7\"\"\"\"#5! \"\"\"\"&F(\"#@F&\"#=" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "The general scheme for multiplying t wo matrices is as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 654 284 284 {PLOTDATA 2 "6J-%'CU RVESG6$7&7$$\"35+++++++?!#=$\"\"!F,7$F+F+7$F+$\"\"'F,7$F(F/-%'COLOURG6 &%$RGBGF,F,F,-F$6$7&7$$\"3q+++++++)*!#F/7$F:F/F2-F$6 $7&7$$\"3$*************>6!#;$!\"$F,7$$\"#6F,FI7$FLF/7$FFF/F2-F$6$7&7$$ \"33++++++!o\"FHFI7$$\"#?FHF+7$$\"#?F,F+7$F\\oF/7$FinF/F2-F$6$7&7$$\"33++++++!e#FHF+7$$\"#EF, F+7$FgoF/7$FdoF/F2-F$6&7$7$$!\"#F,$\"#D!\"\"7$F_pF+7%7$$!++++]=!\"*$\" *++++$FipFdp7$$!++++]@FipFjp-%&STYLEG6#%,PATCHNOGRIDG-F36&F5F+F+$\"*++ ++\"!\")-F$6&7$7$F_p$\"#NFcp7$F_p$\"#gFcp7%7$F]q$\"+++++dFipF^r7$FgpFc rF_qFcq-F$6&7$7$$\"\"$F,$\"\"(F,7$F+F\\s7%7$$\"*+++g$Fip$\"++++]oFipF^ s7$Fas$\"++++]rFipF_qFcq-F$6&7$7$F\\sF\\s7$F>F\\s7%7$$\"++++S'*FipFfsF \\t7$F_tFcsF_qFcq-F$6&7$7$$\"$D\"FcpF\\s7$$\"$5\"FcpF\\s7%7$$\"+++]P6F gqFcsFht7$F]uFfsF_q-%&COLORG6&F5F,$F]sFcpF,-F$6&7$7$$\"$b\"FcpF\\s7$$ \"$q\"FcpF\\s7%7$$\"+++]i;FgqFfsFju7$F_vFcsF_qF`u-F$6&7$7$Fft$\"#:Fcp7 $FftF_r7%7$$\"++++N7Fgq$\"++++ScFipFhv7$$\"++++l7FgqF]wF_qF`u-F$6&7$7$ Fft$\"\"&Fcp7$Fft$!#IFcp7%7$F`w$!++++?FFipFhw7$F[wF]xF_qF`u-F$6&7$7$$ \"#GF,Fap7$FdxF+7%7$$\"++++:GFgqFjpFfx7$$\"++++&y#FgqFjpF_q-Fau6&F5$\" \")FcpF,F`y-F$6&7$7$FdxF\\r7$FdxF_r7%7$F\\yFcrFfy7$FixFcrF_qF^y-F$6&7$ 7$$\"$:#FcpF\\s7$$\"$+#FcpF\\s7%7$$\"+++]P?FgqFcsF`z7$FezFfsF_qF^y-F$6 &7$7$$\"$X#FcpF\\s7$$\"$g#FcpF\\s7%7$$\"+++]iDFgqFfsF^[l7$Fc[lFcsF_qF^ y-F$6%7$7$$\"3))**************HF*$\"\"%F,7$$\"3G*************p*F\"F17$ FGFgqFbqFip-F$6$777$$\"3,++++++L6F[o$\"\"#F77$$\"31+++l[QJ6F[o$\"30+++ 3c(>5#F17$$\"3%******4c(pE6F[o$\"3%******H8pR>#F17$$\"3$******H\"pR>6F [o$\"33+++3c(pE#F17$$\"3-+++hv>56F[o$\"3<+++]'[QJ#F17$$\"#6F7$\"32++++ ++IBF17$$\"3)*******QC!)*3\"F[oFds7$$\"33+++(3.13\"F[oF_s7$$\"33+++RCI t5F[oFjr7$$\"3%******\\8:'o5F[oFer7$$\"3+++++++n5F[oF`r7$Fet$\"3%***** *>RC!)*=F17$Fbt$\"3'******f'3.1=F17$F_t$\"3#******>RCIt\"F17$F\\t$\"3/ +++]8:'o\"F17$Fgs$\"3#*************p;F17$FbsFdu7$F]sFau7$Fhr$\"3/+++n3 .1=F17$FcrF[uF]r-F46&F6$\"#5!\"\"F^qF^q-%%TEXTG6%7$F^qF^qQ#-46\"F3-Fev 6%7$$\"\"\"F7F^qQ\"6FivF3-Fev6%7$F`rF^qQ#-2FivF3-Fev6%7$$\"\"$F7F^qQ\" 1FivF3-Fev6%7$F^qF]wQ\"2FivF3-Fev6%7$F]wF]wQ#-1FivF3-Fev6%7$F`rF]wQ\"4 FivF3-Fev6%7$FgwF]wQ\"3FivF3-Fev6%7$F^qF`rFixF3-Fev6%7$F]wF`rQ\"5FivF3 -Fev6%7$F`rF`rFiwF3-Fev6%7$FgwF`rF]xF3-Fev6%7$$\"#XFcvF`rF]xF3-Fev6%7$ $\"#bFcvF`rFiwF3-Fev6%7$$\"#lFcvF`rFexF3-Fev6%7$$\"#vFcvF`rFixF3-Fev6% 7$$\"#&)FcvF`rFcwF3-Fev6%7$FjyF]wF`yF3-Fev6%7$F_zF]wF]xF3-Fev6%7$FdzF] wFaxF3-Fev6%7$FizF]wQ\"7FivF3-Fev6%7$F^[lF]wFiwF3-Fev6%7$FjyF^qFcwF3-F ev6%7$F_zF^qF_wF3-Fev6%7$FdzF^qFixF3-Fev6%7$FizF^qFexF3-Fev6%7$F^[lF^q F`yF3-Fev6%7$Fjy$FcvF7FixF3-Fev6%7$F_zFb]lFcwF3-Fev6%7$FdzFb]lF`yF3-Fe v6%7$FizFb]lFiwF3-Fev6%7$F^[lFb]lF]xF3-Fev6%7$FgsF`rQ#35FivF3-Fev6%7$$ \"#7F7F`rQ#15FivF3-Fev6%7$$\"#8F7F`rQ#20FivF3-Fev6%7$$\"#9F7F`rQ#50Fiv F3-Fev6%7$$\"#:F7F`rQ\"8FivF3-Fev6%7$FgsF]wQ\"0FivF3-Fev6%7$Ff^lF]wQ#1 8FivF3-Fev6%7$F\\_lF]wQ#36FivF3-Fev6%7$Fb_lF]wFb`lF3-Fev6%7$Fh_lF]wQ#2 1FivF3-Fev6%7$FgsF^qQ#29FivF3-Fev6%7$Ff^lF^qQ#-6FivF3-Fev6%7$F\\_lF^qQ $-23FivF3-Fev6%7$Fb_lF^qQ#23FivF3-Fev6%7$Fh_lF^qF_wF3-Fev6%7$$\"$v*!\" #F]wQ\"=FivF3-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!FivF_cl-%%FONTG6# %(DEFAULTG-%%VIEWG6$FcclFccl" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1 8" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" " Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curv e 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37 " "Curve 38" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "Curve 43" "C urve 44" "Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52" "Curve 53" "Curve 54" "Curve 55" "Curve 56 " "Curve 57" }}}{PARA 0 "" 0 "" {TEXT -1 213 "In the matrix multiplica tion above, the circled entry 35 in the 1st row and 1st column of the \+ product matrix is obtained from the 1st row of the 1st matrix and the \+ 1st column of the 2nd matrix by the calculation: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "``(3)*`.`*``(2)+``(5)*`.`*``(5)+``(1) *`.`*``(-2)+``(2)*`.`*``(3)" "6#,**(-%!G6#\"\"$\"\"\"%\".GF)-F&6#\"\"# F)F)*(-F&6#\"\"&F)F*F)-F&6#F1F)F)*(-F&6#F)F)F*F)-F&6#,$F-!\"\"F)F)*(-F &6#F-F)F*F)-F&6#F(F)F)" }{TEXT -1 3 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "6+25-2+6 = 35;" "6#/,*\"\"'\"\"\"\"#DF&\"\" #!\"\"F%F&\"#N" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 620 161 161 {PLOTDATA 2 "6 hn-%'CURVESG6$7&7$$!3))**************H!#=F(7$$!3++++++++]F*F(7$F,$\"3# )*************H#!#<7$F(F/-%'COLOURG6&%$RGBG\"\"!F7F7-F$6$7&7$$\"3#)*** **********H$F1F(7$$\"3++++++++NF1F(7$F?F/7$F\"F17$FcqFiqFbq Fip-F$6$777$$\"3,++++++L7F[o$\"\"#F77$$\"31+++l[QJ7F[o$\"30+++3c(>5#F1 7$$\"3%******4c(pE7F[o$\"3%******H8pR>#F17$$\"3$******H\"pR>7F[o$\"33+ ++3c(pE#F17$$\"3-+++hv>57F[o$\"3<+++]'[QJ#F17$$\"#7F7$\"32++++++IBF17$ $\"3)*******QC!)*=\"F[oFfs7$$\"33+++(3.1=\"F[oFas7$$\"33+++RCIt6F[oF\\ s7$$\"3%******\\8:'o6F[oFgr7$$\"3+++++++n6F[oFbr7$Fgt$\"3%******>RC!)* =F17$Fdt$\"3'******f'3.1=F17$Fat$\"3#******>RCIt\"F17$F^t$\"3/+++]8:'o \"F17$Fis$\"3#*************p;F17$FdsFfu7$F_sFcu7$Fjr$\"3/+++n3.1=F17$F erF]uF_r-F46&F6$\"#5!\"\"F^qF^q-%%TEXTG6%7$F^qF^qQ#-46\"F3-Fgv6%7$$\" \"\"F7F^qQ\"6F[wF3-Fgv6%7$FbrF^qQ#-2F[wF3-Fgv6%7$$\"\"$F7F^qQ\"1F[wF3- Fgv6%7$F^qF_wQ\"2F[wF3-Fgv6%7$F_wF_wQ#-1F[wF3-Fgv6%7$FbrF_wQ\"4F[wF3-F gv6%7$FiwF_wQ\"3F[wF3-Fgv6%7$F^qFbrF[yF3-Fgv6%7$F_wFbrQ\"5F[wF3-Fgv6%7 $FbrFbrF[xF3-Fgv6%7$FiwFbrF_xF3-Fgv6%7$$\"#XFevFbrF_xF3-Fgv6%7$$\"#bFe vFbrF[xF3-Fgv6%7$$\"#lFevFbrFgxF3-Fgv6%7$$\"#vFevFbrF[yF3-Fgv6%7$$\"#& )FevFbrFewF3-Fgv6%7$F\\zF_wFbyF3-Fgv6%7$FazF_wF_xF3-Fgv6%7$FfzF_wFcxF3 -Fgv6%7$F[[lF_wQ\"7F[wF3-Fgv6%7$F`[lF_wF[xF3-Fgv6%7$F\\zF^qFewF3-Fgv6% 7$FazF^qFawF3-Fgv6%7$FfzF^qF[yF3-Fgv6%7$F[[lF^qFgxF3-Fgv6%7$F`[lF^qFby F3-Fgv6%7$F\\z$FevF7F[yF3-Fgv6%7$FazFd]lFewF3-Fgv6%7$FfzFd]lFbyF3-Fgv6 %7$F[[lFd]lF[xF3-Fgv6%7$F`[lFd]lF_xF3-Fgv6%7$$\"#6F7FbrQ#35F[wF3-Fgv6% 7$FisFbrQ#15F[wF3-Fgv6%7$$\"#8F7FbrQ#20F[wF3-Fgv6%7$$\"#9F7FbrQ#50F[wF 3-Fgv6%7$$\"#:F7FbrQ\"8F[wF3-Fgv6%7$Fd^lF_wQ\"0F[wF3-Fgv6%7$FisF_wQ#18 F[wF3-Fgv6%7$F^_lF_wQ#36F[wF3-Fgv6%7$Fd_lF_wFd`lF3-Fgv6%7$Fj_lF_wQ#21F [wF3-Fgv6%7$Fd^lF^qQ#29F[wF3-Fgv6%7$FisF^qQ#-6F[wF3-Fgv6%7$F^_lF^qQ$-2 3F[wF3-Fgv6%7$Fd_lF^qQ#23F[wF3-Fgv6%7$Fj_lF^qFawF3-Fgv6%7$$\"$v*!\"#F_ wQ\"=F[wF3-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!F[wFacl-%%FONTG6#%(D EFAULTG-%%VIEWG6$FeclFecl" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+ 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Cur ve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 3 8" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "Curve 43" "Curve 44" " Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curv e 51" "Curve 52" "Curve 53" "Curve 54" "Curve 55" "Curve 56" "Curve 57 " }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 177 "The circled entry 1 5 in the 1st row and 2nd column of the product matrix is obtained from the 1st row of the 1st matrix and the 2nd column of the 2nd matrix by the calculation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(3)*`.`*``(1)+``(5)*`.`*``(2)+``(1)*`.`*``(6)+``(2)*`.`*``(-2);" "6 #,**(-%!G6#\"\"$\"\"\"%\".GF)-F&6#F)F)F)*(-F&6#\"\"&F)F*F)-F&6#\"\"#F) F)*(-F&6#F)F)F*F)-F&6#\"\"'F)F)*(-F&6#F3F)F*F)-F&6#,$F3!\"\"F)F)" } {TEXT -1 3 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 " 3+10+6-4=15" "6#/,*\"\"$\"\"\"\"#5F&\"\"'F&\"\"%!\"\"\"#:" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 643 160 160 {PLOTDATA 2 "6hn-%'CURVESG6$7&7$$!3))****** ********H!#=F(7$$!3++++++++]F*F(7$F,$\"3#)*************H#!#<7$F(F/-%'C OLOURG6&%$RGBG\"\"!F7F7-F$6$7&7$$\"3#)*************H$F1F(7$$\"3+++++++ +NF1F(7$F?F/7$F\"F17$$\"3;+++++++KF1Fcp7$Ffp F`pF_p-F46&F6$\"*++++\"!\")$F7F7F^q-F$6$7'7$$\"3;+++++++sF1$\"3;++++++ +AF17$$\"3;+++++++xF1Feq7$Fhq$!3%**************>\"F17$FcqF[rFbqFip-F$6 $777$$\"3,++++++L9F[o$\"\"\"F77$$\"31+++l[QJ9F[o$\"30+++3c(>5\"F17$$\" 3%******4c(pE9F[o$\"3%******H8pR>\"F17$$\"3$******H\"pR>9F[o$\"33+++3c (pE\"F17$$\"3-+++hv>59F[o$\"3&*******\\'[QJ\"F17$$\"#9F7$\"32++++++I8F 17$$\"3)*******QC!)*Q\"F[oFhs7$$\"33+++(3.1Q\"F[oFcs7$$\"33+++RCIt8F[o F^s7$$\"3%******\\8:'o8F[oFir7$$\"3+++++++n8F[o$\"2+++!**********F17$F it$\"3W+++=RC!)*)F*7$Fft$\"3)******\\m3.1)F*7$Fct$\"33+++=RCItF*7$F`t$ \"33+++&\\8:'oF*7$F[t$\"3S+++++++nF*7$Ffs$\"3C+++(\\8:'oF*7$Fas$\"3D++ +?RCItF*7$F\\s$\"3K+++p'3.1)F*7$Fgr$\"3m*****>#RC!)*)F*Far-F46&F6$\"#5 !\"\"F^qF^q-%%TEXTG6%7$F^qF^qQ#-46\"F3-Faw6%7$FdrF^qQ\"6FewF3-Faw6%7$$ \"\"#F7F^qQ#-2FewF3-Faw6%7$$\"\"$F7F^qQ\"1FewF3-Faw6%7$F^qFdrQ\"2FewF3 -Faw6%7$FdrFdrQ#-1FewF3-Faw6%7$F]xFdrQ\"4FewF3-Faw6%7$FcxFdrQ\"3FewF3- Faw6%7$F^qF]xFeyF3-Faw6%7$FdrF]xQ\"5FewF3-Faw6%7$F]xF]xFexF3-Faw6%7$Fc xF]xFixF3-Faw6%7$$\"#XF_wF]xFixF3-Faw6%7$$\"#bF_wF]xFexF3-Faw6%7$$\"#l F_wF]xFayF3-Faw6%7$$\"#vF_wF]xFeyF3-Faw6%7$$\"#&)F_wF]xF_xF3-Faw6%7$Ff zFdrF\\zF3-Faw6%7$F[[lFdrFixF3-Faw6%7$F`[lFdrF]yF3-Faw6%7$Fe[lFdrQ\"7F ewF3-Faw6%7$Fj[lFdrFexF3-Faw6%7$FfzF^qF_xF3-Faw6%7$F[[lF^qFiwF3-Faw6%7 $F`[lF^qFeyF3-Faw6%7$Fe[lF^qFayF3-Faw6%7$Fj[lF^qF\\zF3-Faw6%7$Ffz$F_wF 7FeyF3-Faw6%7$F[[lF^^lF_xF3-Faw6%7$F`[lF^^lF\\zF3-Faw6%7$Fe[lF^^lFexF3 -Faw6%7$Fj[lF^^lFixF3-Faw6%7$$\"#6F7F]xQ#35FewF3-Faw6%7$$\"#7F7F]xQ#15 FewF3-Faw6%7$$\"#8F7F]xQ#20FewF3-Faw6%7$F[tF]xQ#50FewF3-Faw6%7$$\"#:F7 F]xQ\"8FewF3-Faw6%7$F^_lFdrQ\"0FewF3-Faw6%7$Fd_lFdrQ#18FewF3-Faw6%7$Fj _lFdrQ#36FewF3-Faw6%7$F[tFdrF^alF3-Faw6%7$Fd`lFdrQ#21FewF3-Faw6%7$F^_l F^qQ#29FewF3-Faw6%7$Fd_lF^qQ#-6FewF3-Faw6%7$Fj_lF^qQ$-23FewF3-Faw6%7$F [tF^qQ#23FewF3-Faw6%7$Fd`lF^qFiwF3-Faw6%7$$\"$v*!\"#FdrQ\"=FewF3-%*AXE SSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!FewF[dl-%%FONTG6#%(DEFAULTG-%%VIEWG6 $F_dlF_dl" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Cu rve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Cu rve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14 " "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "C urve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33 " "Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" "C urve 40" "Curve 41" "Curve 42" "Curve 43" "Curve 44" "Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52 " "Curve 53" "Curve 54" "Curve 55" "Curve 56" "Curve 57" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 177 "The circled entry 18 in the 2nd ro w and 4th column of the product matrix is obtained from the 2nd row of the 1st matrix and the 4th column of the 2nd matrix by the calculatio n: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(2)*`.`*``(3 )+``(-1)*`.`*``(7)+``(4)*`.`*``(4)+``(3)*`.`*``(1);" "6#,**(-%!G6#\"\" #\"\"\"%\".GF)-F&6#\"\"$F)F)*(-F&6#,$F)!\"\"F)F*F)-F&6#\"\"(F)F)*(-F&6 #\"\"%F)F*F)-F&6#F9F)F)*(-F&6#F-F)F*F)-F&6#F)F)F)" }{TEXT -1 3 " " } }{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "6-7+16+3=18" "6#/, *\"\"'\"\"\"\"\"(!\"\"\"#;F&\"\"$F&\"#=" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "This calculation c an be performed with Maple as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "A := Matrix([[3,5,1,2] ,[2,-1,4,3],[-4,6,-2,1]]);\nB := Matrix([[2,1,4,3,-2],[5,2,-1,7,1],[-2 ,6,3,4,5],[3,-2,5,1,2]]);\n'A.B'=A.B;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\"*'fha9-%'MATRIXG6#7%7&\"\"$\"\"&\"\"\"\"\"#7&F1 !\"\"\"\"%F.7&!\"%\"\"'!\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"B G-%'RTABLEG6$\"*GNYX\"-%'MATRIXG6#7&7'\"\"#\"\"\"\"\"%\"\"$!\"#7'\"\"& F.!\"\"\"\"(F/7'F2\"\"'F1F0F47'F1F2F4F/F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\".G6$%\"AG%\"BG-%'RTABLEG6$\"*o^YX\"-%'MATRIXG6#7%7 '\"#N\"#:\"#?\"#]\"\")7'\"\"!\"#=\"#OF9\"#@7'\"#H!\"'!#B\"#B\"\"'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "In general, if " }{XPPEDIT 18 0 "A = [a[i,j]][m,p];" "6#/%\"AG&7#&%\"aG6$%\"iG%\"jG6$%\"mG%\"pG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "B = [b[i,j]][p, n];" "6#/%\"BG&7#&% \"bG6$%\"iG%\"jG6$%\"pG%\"nG" }{TEXT -1 8 ", then " }{XPPEDIT 18 0 "A *`.`*B = [c[i,j]][m,n];" "6#/*(%\"AG\"\"\"%\".GF&%\"BGF&&7#&%\"cG6$%\" iG%\"jG6$%\"mG%\"nG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "c[i,j]=Sum (a[i,k]*b[k,j],k=1..p)" "6#/&%\"cG6$%\"iG%\"jG-%$SumG6$*&&%\"aG6$F'%\" kG\"\"\"&%\"bG6$F0F(F1/F0;F1%\"pG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 14 "Other examples" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "matrix([ [4,-1,2],[3,7,6]])" "6#-%'matrixG6#7$7%\"\"%,$\"\"\"!\"\"\"\"#7%\"\"$ \"\"(\"\"'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[-2,1],[3,-4],[2,7 ]])=matrix([[-7, 22], [27, 17]])" "6#/-%'matrixG6#7%7$,$\"\"#!\"\"\"\" \"7$\"\"$,$\"\"%F+7$F*\"\"(-F%6#7$7$,$F2F+\"#A7$\"#F\"#<" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 52 "The entry -7 in the product matrix is obtained from " }{XPPEDIT 18 0 "``(4)*`.`*``(-2)+``(-1)*`.`*``(3)+ ``(2)*`.`*``(2)=-8-3+4" "6#/,(*(-%!G6#\"\"%\"\"\"%\".GF*-F'6#,$\"\"#! \"\"F*F**(-F'6#,$F*F0F*F+F*-F'6#\"\"$F*F**(-F'6#F/F*F+F*-F'6#F/F*F*,( \"\")F0F7F0F)F*" }{XPPEDIT 18 0 "``=-7" "6#/%!G,$\"\"(!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 52 "The entry 22 in the product m atrix is obtained from " }{XPPEDIT 18 0 "``(4)*`.`*``(1)+``(-1)*`.`*`` (-4)+``(2)*`.`*``(7) = 4+4+14;" "6#/,(*(-%!G6#\"\"%\"\"\"%\".GF*-F'6#F *F*F**(-F'6#,$F*!\"\"F*F+F*-F'6#,$F)F2F*F**(-F'6#\"\"#F*F+F*-F'6#\"\"( F*F*,(F)F*F)F*\"#9F*" }{XPPEDIT 18 0 "`` = 22;" "6#/%!G\"#A" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 52 "The entry 27 in the product m atrix is obtained from " }{XPPEDIT 18 0 "``(3)*`.`*``(-2)+``(7)*`.`*`` (3)+``(6)*`.`*``(2) = -6+21+12;" "6#/,(*(-%!G6#\"\"$\"\"\"%\".GF*-F'6# ,$\"\"#!\"\"F*F**(-F'6#\"\"(F*F+F*-F'6#F)F*F**(-F'6#\"\"'F*F+F*-F'6#F/ F*F*,(F:F0\"#@F*\"#7F*" }{XPPEDIT 18 0 "`` = 27;" "6#/%!G\"#F" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 52 "The entry 17 in the product m atrix is obtained from " }{XPPEDIT 18 0 "``(3)*`.`*``(1)+``(7)*`.`*``( -4)+``(6)*`.`*``(7) = 3-28+42;" "6#/,(*(-%!G6#\"\"$\"\"\"%\".GF*-F'6#F *F*F**(-F'6#\"\"(F*F+F*-F'6#,$\"\"%!\"\"F*F**(-F'6#\"\"'F*F+F*-F'6#F1F *F*,(F)F*\"#GF6\"#UF*" }{XPPEDIT 18 0 "`` = 17;" "6#/%!G\"#<" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1,2], [3,4]])" "6#-%'matrixG6#7$7$\"\"\"\"\"#7$\"\"$\"\"%" }{TEXT -1 2 " " }{XPPEDIT 18 0 "matrix([[5,6],[7,8]])=matrix([[19,22],[43,50]])" "6#/- %'matrixG6#7$7$\"\"&\"\"'7$\"\"(\"\")-F%6#7$7$\"#>\"#A7$\"#V\"#]" } {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[3, -4, -5], [-5, 3, 7], [2, 7, 1]])" "6#-%'matrixG6#7%7%\"\"$,$\"\"%!\"\",$\"\"&F+7%,$F-F+F(\"\"(7% \"\"#F0\"\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "matrix([[-3, 4, 3], [0 , 4, -7], [4, 3, -4]])=matrix([[-29, -19, 57], [43, 13, -64], [-2, 39, -47]])" "6#/-%'matrixG6#7%7%,$\"\"$!\"\"\"\"%F*7%\"\"!F,,$\"\"(F+7%F, F*,$F,F+-F%6#7%7%,$\"#HF+,$\"#>F+\"#d7%\"#V\"#8,$\"#kF+7%,$\"\"#F+\"#R ,$\"#ZF+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 58 "The number of rows and columns of a matrix constit ute its " }{TEXT 259 10 "dimensions" }{TEXT -1 238 ". Notice that the \+ product of two square matrices with the same dimensions, or dimension \+ (with no \"s\", since the number of rows is the same as the number of \+ columns), is a matrix of the same dimension as each of the original tw o matrices. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "Propert ies of matrix multiplication " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 42 "The operation of matrix multip lication is " }{TEXT 259 15 "not commutative" }{TEXT -1 28 ", that is, for two matrices " }{TEXT 287 1 "A" }{TEXT -1 5 " and " }{TEXT 286 1 "B" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "A*B<>B*A" "6#0*&%\"AG\"\"\"%\"BGF &*&F'F&F%F&" }{TEXT -1 14 ", in general. " }}{PARA 0 "" 0 "" {TEXT -1 130 " In fact, reversing the order of the two matrices in a product ma y lead to a situation where the new product is not even defined. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "For examp le, " }{XPPEDIT 18 0 "matrix([[3, 5]])*matrix([[2, 7], [4, 6]]) = mat rix([[26, 51]]);" "6#/*&-%'matrixG6#7#7$\"\"$\"\"&\"\"\"-F&6#7$7$\"\"# \"\"(7$\"\"%\"\"'F,-F&6#7#7$\"#E\"#^" }{TEXT -1 21 ", while the produc t " }{XPPEDIT 18 0 "matrix([[2, 7], [4, 6]])*matrix([[3, 5]]);" "6#*& -%'matrixG6#7$7$\"\"#\"\"(7$\"\"%\"\"'\"\"\"-F%6#7#7$\"\"$\"\"&F." } {TEXT -1 17 " is not defined. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 76 "Even when both products are defined, they may not have the same dimensions. " }}{PARA 0 "" 0 "" {TEXT -1 15 "Fo r example, " }{XPPEDIT 18 0 "matrix([[1, 2]])*matrix([[3], [4]]) = m atrix([[11]]);" "6#/*&-%'matrixG6#7#7$\"\"\"\"\"#F*-F&6#7$7#\"\"$7#\" \"%F*-F&6#7#7#\"#6" }{TEXT -1 10 ", while " }{XPPEDIT 18 0 "matrix([ [3], [4]])*matrix([[1, 2]]) = matrix([[3, 6], [4, 8]]);" "6#/*&-%'matr ixG6#7$7#\"\"$7#\"\"%\"\"\"-F&6#7#7$F-\"\"#F--F&6#7$7$F*\"\"'7$F,\"\") " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "If we restrict attention to square matrices " }{TEXT 284 1 "A" }{TEXT -1 5 " and " }{TEXT 285 1 "B" }{TEXT -1 29 ", then both o f the products " }{XPPEDIT 18 0 "A*B;" "6#*&%\"AG\"\"\"%\"BGF%" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "B*A;" "6#*&%\"BG\"\"\"%\"AGF%" } {TEXT -1 21 " are always defined. " }}{PARA 0 "" 0 "" {TEXT -1 21 "The product of two 2 " }{TEXT 283 1 "x" }{TEXT -1 25 " 2 matrices is give n by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[a, b],[c,d]])" "6#-%'matrixG6#7$7$%\"aG%\"bG7$%\"cG%\"dG" }{TEXT -1 2 " \+ " }{XPPEDIT 18 0 "matrix([[p,q],[r,s]])=matrix([[a*p+b*r,a*q+b*s],[c*p +d*r,c*q+d*s]])" "6#/-%'matrixG6#7$7$%\"pG%\"qG7$%\"rG%\"sG-F%6#7$7$,& *&%\"aG\"\"\"F)F5F5*&%\"bGF5F,F5F5,&*&F4F5F*F5F5*&F7F5F-F5F57$,&*&%\"c GF5F)F5F5*&%\"dGF5F,F5F5,&*&F>F5F*F5F5*&F@F5F-F5F5" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "A=matrix([[1,2],[3, 4]])" "6#/%\"AG-%'matrixG6#7$7$\"\"\"\"\"#7$\"\"$\"\"%" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "B=matrix([[-2,1],[5,3])" "6#/%\"BG-%'matrixG6# 7$7$,$\"\"#!\"\"\"\"\"7$\"\"&\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 6 "Then " }{XPPEDIT 18 0 "A*B = matrix([[1,2],[3,4]])*matr ix([[-2,1],[5,3])" "6#/*&%\"AG\"\"\"%\"BGF&*&-%'matrixG6#7$7$F&\"\"#7$ \"\"$\"\"%F&-F*6#7$7$,$F.!\"\"F&7$\"\"&F0F&" }{XPPEDIT 18 0 "`` = matr ix([[8, 7], [14, 15]]);" "6#/%!G-%'matrixG6#7$7$\"\")\"\"(7$\"#9\"#:" }{TEXT -1 8 ", while " }{XPPEDIT 18 0 "B*A=matrix([[-2,1],[5,3])*matri x([[1,2],[3,4]])" "6#/*&%\"BG\"\"\"%\"AGF&*&-%'matrixG6#7$7$,$\"\"#!\" \"F&7$\"\"&\"\"$F&-F*6#7$7$F&F/7$F3\"\"%F&" }{XPPEDIT 18 0 "`` = matri x([[1, 0], [14, 22]]);" "6#/%!G-%'matrixG6#7$7$\"\"\"\"\"!7$\"#9\"#A" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "A := Matrix( [[1,2],[3,4]]);\nB := Matrix([[-2,1],[5,3]]);\n'A.B'=A.B;\n'B.A'=B.A; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\"*SzYX\"-%'MATR IXG6#7$7$\"\"\"\"\"#7$\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"BG-%'RTABLEG6$\"*36ZX\"-%'MATRIXG6#7$7$!\"#\"\"\"7$\"\"&\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\".G6$%\"AG%\"BG-%'RTABLEG6$\"*3QZX \"-%'MATRIXG6#7$7$\"\")\"\"(7$\"#9\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\".G6$%\"BG%\"AG-%'RTABLEG6$\"*wyZX\"-%'MATRIXG6#7$7$\"\"\"\" \"!7$\"#9\"#A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "On the ot her hand, given three matrices " }{TEXT 319 1 "A" }{TEXT -1 2 ", " } {TEXT 320 1 "B" }{TEXT -1 5 " and " }{TEXT 321 1 "C" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A*``(B*C) = ``(A*B) *C;" "6#/*&%\"AG\"\"\"-%!G6#*&%\"BGF&%\"CGF&F&*&-F(6#*&F%F&F+F&F&F,F& " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "whenever, the product s are defined." }}{PARA 0 "" 0 "" {TEXT -1 30 "Thus matrix multiplicat ion is " }{TEXT 259 11 "associative" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 18 "For example, let " }{XPPEDIT 18 0 "A=matrix([[1,2],[3, 4]])" "6#/%\"AG-%'matrixG6#7$7$\"\"\"\"\"#7$\"\"$\"\"%" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "B=matrix([[-2,1],[5,3])" "6#/%\"BG-%'matrixG6# 7$7$,$\"\"#!\"\"\"\"\"7$\"\"&\"\"$" }{TEXT -1 21 ", as before, and let " }{XPPEDIT 18 0 "C=matrix([[3,7],[4,1]])" "6#/%\"CG-%'matrixG6#7$7$ \"\"$\"\"(7$\"\"%\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "A*B = matrix([[1,2],[3,4]])*matrix([[-2,1],[ 5,3])" "6#/*&%\"AG\"\"\"%\"BGF&*&-%'matrixG6#7$7$F&\"\"#7$\"\"$\"\"%F& -F*6#7$7$,$F.!\"\"F&7$\"\"&F0F&" }{XPPEDIT 18 0 "`` = matrix([[8, 7], \+ [14, 15]]);" "6#/%!G-%'matrixG6#7$7$\"\")\"\"(7$\"#9\"#:" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "``(A*B)*C = matrix([[8, 7], [14, 15]])*matrix( [[3, 7], [4, 1]]);" "6#/*&-%!G6#*&%\"AG\"\"\"%\"BGF*F*%\"CGF**&-%'matr ixG6#7$7$\"\")\"\"(7$\"#9\"#:F*-F/6#7$7$\"\"$F47$\"\"%F*F*" }{XPPEDIT 18 0 "``= matrix([[52, 63], [102, 113]])" "6#/%!G-%'matrixG6#7$7$\"#_ \"#j7$\"$-\"\"$8\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 11 "Similarly, " }{XPPEDIT 18 0 "B*C=matrix([ [-2, 1], [5, 3]])*matrix([[3, 7], [4, 1]])" "6#/*&%\"BG\"\"\"%\"CGF&*& -%'matrixG6#7$7$,$\"\"#!\"\"F&7$\"\"&\"\"$F&-F*6#7$7$F3\"\"(7$\"\"%F&F &" }{XPPEDIT 18 0 "``=matrix([[-2, -13], [27, 38]])" "6#/%!G-%'matrixG 6#7$7$,$\"\"#!\"\",$\"#8F,7$\"#F\"#Q" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "A*``(B*C) =matrix([[1,2], [3,4])*matrix([[-2, -13], [27, 38]])" "6#/*&%\"AG\"\"\"-%!G6#*&%\"BGF&%\"CGF&F&*&-%'matrixG6#7$7$F&\"\"#7$\" \"$\"\"%F&-F/6#7$7$,$F3!\"\",$\"#8F<7$\"#F\"#QF&" }{XPPEDIT 18 0 "``= \+ matrix([[52, 63], [102, 113]])" "6#/%!G-%'matrixG6#7$7$\"#_\"#j7$\"$- \"\"$8\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "A := Matrix([[1,2],[3,4]]):\nB := Matrix([[-2,1],[5,3]]):\nC := Matrix([[3 ,7],[4,1]]):\n'A'=A,'B'=B,'C'=C;\n'A*B'=A.B;\n`(`*'A*B'*`)`*'C'=A.B.C; \n'B*C'=B.C;\n'A'*`(`*'B*C'*`)`=A.B.C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%\"AG-%'RTABLEG6$\"*[1[X\"-%'MATRIXG6#7$7$\"\"\"\"\"#7$\"\"$\" \"%/%\"BG-F&6$\"*_?[X\"-F*6#7$7$!\"#F.7$\"\"&F1/%\"CG-F&6$\"*_Z[X\"-F* 6#7$7$F1\"\"(7$F2F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\"\"% \"BGF&-%'RTABLEG6$\"*Wk[X\"-%'MATRIXG6#7$7$\"\")\"\"(7$\"#9\"#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"(G\"\"\"%\"AGF&%\"BGF&%\")GF&%\" CGF&-%'RTABLEG6$\"*;#*[X\"-%'MATRIXG6#7$7$\"#_\"#j7$\"$-\"\"$8\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"BG\"\"\"%\"CGF&-%'RTABLEG6$\"*)= !\\X\"-%'MATRIXG6#7$7$!\"#!#87$\"#F\"#Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"(G\"\"\"%\"AGF&%\"BGF&%\")GF&%\"CGF&-%'RTABLEG6$\"*KI\\X\" -%'MATRIXG6#7$7$\"#_\"#j7$\"$-\"\"$8\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Matrix multiplication is " }{TEXT 259 26 "distributive ov er addition" }{TEXT -1 46 ", both on the left and on the right, that i s, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A*(B+C)=A*B+A* C" "6#/*&%\"AG\"\"\",&%\"BGF&%\"CGF&F&,&*&F%F&F(F&F&*&F%F&F)F&F&" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "(A+B)*C=A*C+B*C" "6#/*&,&%\"AG\"\"\"% \"BGF'F'%\"CGF',&*&F&F'F)F'F'*&F(F'F)F'F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 35 "whenever the products are defined. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "For example, let \+ " }{XPPEDIT 18 0 "A=matrix([[1,2],[3,4]])" "6#/%\"AG-%'matrixG6#7$7$\" \"\"\"\"#7$\"\"$\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "B=matrix([[-2 ,1],[5,3])" "6#/%\"BG-%'matrixG6#7$7$,$\"\"#!\"\"\"\"\"7$\"\"&\"\"$" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "C=matrix([[3,7],[4,1]])" "6#/%\"CG- %'matrixG6#7$7$\"\"$\"\"(7$\"\"%\"\"\"" }{TEXT -1 13 ", as before. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " } {XPPEDIT 18 0 "B+C=matrix([[1, 8], [9, 4]])" "6#/,&%\"BG\"\"\"%\"CGF&- %'matrixG6#7$7$F&\"\")7$\"\"*\"\"%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "A*(B+C)= matrix([[1, 2], [3, 4]])*matrix([[1, 8], [9, 4]])" "6#/*&% \"AG\"\"\",&%\"BGF&%\"CGF&F&*&-%'matrixG6#7$7$F&\"\"#7$\"\"$\"\"%F&-F, 6#7$7$F&\"\")7$\"\"*F3F&" }{XPPEDIT 18 0 "``=matrix([[19, 16], [39, 40 ]])" "6#/%!G-%'matrixG6#7$7$\"#>\"#;7$\"#R\"#S" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Also, " } {XPPEDIT 18 0 "A*B+A*C= matrix([[1, 2], [3, 4]])*matrix([[-2, 1], [5, \+ 3]])+matrix([[1, 2], [3, 4]])*matrix([[3, 7], [4, 1]])" "6#/,&*&%\"AG \"\"\"%\"BGF'F'*&F&F'%\"CGF'F',&*&-%'matrixG6#7$7$F'\"\"#7$\"\"$\"\"%F '-F.6#7$7$,$F2!\"\"F'7$\"\"&F4F'F'*&-F.6#7$7$F'F27$F4F5F'-F.6#7$7$F4\" \"(7$F5F'F'F'" }{TEXT -1 4 " = " }{XPPEDIT 18 0 " matrix([[8, 7], [14 , 15]])+matrix([[11, 9], [25, 25]])= matrix([[19, 16], [39, 40]])" "6# /,&-%'matrixG6#7$7$\"\")\"\"(7$\"#9\"#:\"\"\"-F&6#7$7$\"#6\"\"*7$\"#DF 7F/-F&6#7$7$\"#>\"#;7$\"#R\"#S" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "This illustrates the lef t hand distributive law. The right hand distributive law can be illust rated in a similar way. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "A := Matrix([[1,2],[3,4]]):\nB := \+ Matrix([[-2,1],[5,3]]):\nC := Matrix([[3,7],[4,1]]):\n'A'=A,'B'=B,'C'= C;\nconvert(A.C,matrix);\n'B+C'=B+C;\n'A*(B+C)'=A.(B+C);\n'A*B'=A.B;\n 'A*C'=A.C;\n'A*B+A*C'=A.B+A.C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%\" AG-%'RTABLEG6$\"*!e%\\X\"-%'MATRIXG6#7$7$\"\"\"\"\"#7$\"\"$\"\"%/%\"BG -F&6$\"*s*)\\X\"-F*6#7$7$!\"#F.7$\"\"&F1/%\"CG-F&6$\"*[/]X\"-F*6#7$7$F 1\"\"(7$F2F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"#6\" \"*7$\"#DF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"BG\"\"\"%\"CGF&-% 'RTABLEG6$\"*Sd]X\"-%'MATRIXG6#7$7$F&\"\")7$\"\"*\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\"\",&%\"BGF&%\"CGF&F&-%'RTABLEG6$\"*;s] X\"-%'MATRIXG6#7$7$\"#>\"#;7$\"#R\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\"\"%\"BGF&-%'RTABLEG6$\"*;*4b9-%'MATRIXG6#7$7$\"\")\"\"( 7$\"#9\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\"\"%\"CGF&-%' RTABLEG6$\"*?8^X\"-%'MATRIXG6#7$7$\"#6\"\"*7$\"#DF4" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,&*&%\"AG\"\"\"%\"BGF'F'*&F&F'%\"CGF'F'-%'RTABLEG6$ \"*?S^X\"-%'MATRIXG6#7$7$\"#>\"#;7$\"#R\"#S" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "The left hand distributive law \+ can be \"proved\" for 2 " }{TEXT 288 1 "x" }{TEXT -1 24 " 2 matrices a s follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 272 "unassign('a','b','c','d','p','q','r','s','t','u',' v','w'):\nA := Matrix([[a,b],[c,d]]):\nB := Matrix([[p,q],[r,s]]):\nC \+ := Matrix([[t,u],[v,w]]):\n'A'=A,'B'=B,'C'=C;\n'B+C'=B+C;\n'A*(B+C)'=A .(B+C);\n``=map(expand,convert(A.(B+C),matrix));\n'A*B'=A.B;\n'A*C'=A. C;\n'A*B+A*C'=A.B+A.C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%\"AG-%'RTA BLEG6$\"*Ca^X\"-%'MATRIXG6#7$7$%\"aG%\"bG7$%\"cG%\"dG/%\"BG-F&6$\"*'>= b9-F*6#7$7$%\"pG%\"qG7$%\"rG%\"sG/%\"CG-F&6$\"*#H@b9-F*6#7$7$%\"tG%\"u G7$%\"vG%\"wG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"BG\"\"\"%\"CGF& -%'RTABLEG6$\"*!GCb9-%'MATRIXG6#7$7$,&%\"pGF&%\"tGF&,&%\"qGF&%\"uGF&7$ ,&%\"rGF&%\"vGF&,&%\"sGF&%\"wGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/* &%\"AG\"\"\",&%\"BGF&%\"CGF&F&-%'RTABLEG6$\"*[$Gb9-%'MATRIXG6#7$7$,&*& %\"aGF&,&%\"pGF&%\"tGF&F&F&*&%\"bGF&,&%\"rGF&%\"vGF&F&F&,&*&F5F&,&%\"q GF&%\"uGF&F&F&*&F:F&,&%\"sGF&%\"wGF&F&F&7$,&*&%\"cGF&F6F&F&*&%\"dGF&F; F&F&,&*&FJF&F@F&F&*&FLF&FDF&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G -%'matrixG6#7$7$,**&%\"aG\"\"\"%\"pGF-F-*&%\"bGF-%\"rGF-F-*&F,F-%\"tGF -F-*&F0F-%\"vGF-F-,**&F,F-%\"qGF-F-*&F0F-%\"sGF-F-*&F,F-%\"uGF-F-*&F0F -%\"wGF-F-7$,**&%\"cGF-F.F-F-*&%\"dGF-F1F-F-*&FBF-F3F-F-*&FDF-F5F-F-,* *&FBF-F8F-F-*&FDF-F:F-F-*&FBF-FF-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\"\"%\"BGF&-%'RTABLEG6$\"*#*H`X\"-%'MATRIXG6# 7$7$,&*&%\"aGF&%\"pGF&F&*&%\"bGF&%\"rGF&F&,&*&F3F&%\"qGF&F&*&F6F&%\"sG F&F&7$,&*&%\"cGF&F4F&F&*&%\"dGF&F7F&F&,&*&F@F&F:F&F&*&FBF&FF'F',**&FKF'FAF'F'*&FMF'FCF'F'*&FKF'FEF'F'* &FMF'FGF'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "The ide ntity matrix and inverse matrices " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 11 "The matrix " } {XPPEDIT 18 0 "I=matrix([[1,0],[0,1]])" "6#/%\"IG-%'matrixG6#7$7$\"\" \"\"\"!7$F+F*" }{TEXT -1 16 " is called the " }{TEXT 259 8 "identity " }{TEXT -1 4 " or " }{TEXT 259 4 "unit" }{TEXT -1 3 " 2 " }{TEXT 289 1 "x" }{TEXT -1 11 " 2 matrix. " }}{PARA 0 "" 0 "" {TEXT -1 26 "It has the property that " }{XPPEDIT 18 0 "I*`.`*A = A;" "6#/*(%\"IG\"\"\"% \".GF&%\"AGF&F(" }{TEXT -1 17 " for any matrix " }{TEXT 291 1 "A" } {TEXT -1 26 " which has two rows, and " }{XPPEDIT 18 0 "B*`.`*I=B" "6 #/*(%\"BG\"\"\"%\".GF&%\"IGF&F%" }{TEXT -1 17 " for any matrix " } {TEXT 290 1 "B" }{TEXT -1 22 " which has 2 columns. " }}{PARA 0 "" 0 " " {TEXT -1 13 "For example, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[1,0],[0,1]])*matrix([[1,2,3],[4,5,6]])=matrix( [[1,2,3],[4,5,6]])" "6#/*&-%'matrixG6#7$7$\"\"\"\"\"!7$F+F*F*-F&6#7$7% F*\"\"#\"\"$7%\"\"%\"\"&\"\"'F*-F&6#7$7%F*F1F27%F4F5F6" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1,2],[3,4],[5,6]])*matrix([[1,0],[0,1]])=m atrix([[1,2],[3,4],[5,6]])" "6#/*&-%'matrixG6#7%7$\"\"\"\"\"#7$\"\"$\" \"%7$\"\"&\"\"'F*-F&6#7$7$F*\"\"!7$F6F*F*-F&6#7%7$F*F+7$F-F.7$F0F1" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "For any 2 " }{TEXT 292 1 "x" }{TEXT -1 10 " 2 matrix " } {TEXT 318 1 "A" }{TEXT -1 10 " we have " }{XPPEDIT 18 0 "I*`.`*A=A" " 6#/*(%\"IG\"\"\"%\".GF&%\"AGF&F(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "A*`.`*I=A" "6#/*(%\"AG\"\"\"%\".GF&%\"IGF&F%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 14 "For example, " }{XPPEDIT 18 0 "matrix([[ 1, 0], [0, 1]])*matrix([[5,2], [3,-7]]) = matrix([[5,2], [3,-7]])" "6# /*&-%'matrixG6#7$7$\"\"\"\"\"!7$F+F*F*-F&6#7$7$\"\"&\"\"#7$\"\"$,$\"\" (!\"\"F*-F&6#7$7$F1F27$F4,$F6F7" }{TEXT -1 9 " and " }{XPPEDIT 18 0 "matrix([[5,2], [3,-7]])*matrix([[1, 0], [0, 1]]) = matrix([[5,2], [ 3,-7]])" "6#/*&-%'matrixG6#7$7$\"\"&\"\"#7$\"\"$,$\"\"(!\"\"\"\"\"-F&6 #7$7$F1\"\"!7$F6F1F1-F&6#7$7$F*F+7$F-,$F/F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The unit 3 x 3 \+ matrix is " }{XPPEDIT 18 0 "I=`` " "6#/%\"IG%!G" }{XPPEDIT 18 0 "I[3] = matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]]);" "6#/&%\"IG6#\"\"$-%'mat rixG6#7%7%\"\"\"\"\"!F.7%F.F-F.7%F.F.F-" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "The unit " }{TEXT 297 1 "n" }{TEXT -1 1 " " }{TEXT 299 1 "x" }{TEXT -1 1 " " }{TEXT 298 1 "n" }{TEXT -1 11 " matrix is " }{XPPEDIT 18 0 "I=``" "6#/%\"IG%!G" } {XPPEDIT 18 0 "I[n] =matrix([[1,0,0,` . . . `,0],[0,1,0,` . . . `,0],[ 0,0,1,` . . . `,0],[``,``,``,``,``],[0,0,0,` . . . `,1]])" "6#/&%\"IG6 #%\"nG-%'matrixG6#7'7'\"\"\"\"\"!F.%(~.~.~.~GF.7'F.F-F.F/F.7'F.F.F-F/F .7'%!GF3F3F3F37'F.F.F.F/F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "The unit matrix " }{TEXT 303 1 "I" }{TEXT -1 58 " has 1's along the main diagonal and 0's everywhere \+ else. " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 5 ": The" } {TEXT 259 14 " main diagonal" }{TEXT -1 8 " of an " }{TEXT 300 1 "n" }{TEXT -1 1 " " }{TEXT 302 1 "x" }{TEXT -1 1 " " }{TEXT 301 1 "n" } {TEXT -1 16 " square matrix " }{XPPEDIT 18 0 "A=matrix([[a[1,1], a[1, 2], ` . . . `, a[1,n]], [a[2,1], a[2,2], ` . . . `, a[n,1]], [``, ``, \+ ``, ``], [a[n,1], a[n,2], ` . . . `, a[n.n]]])" "6#/%\"AG-%'matrixG6#7 &7&&%\"aG6$\"\"\"F-&F+6$F-\"\"#%(~.~.~.~G&F+6$F-%\"nG7&&F+6$F0F-&F+6$F 0F0F1&F+6$F4F-7&%!GF=F=F=7&&F+6$F4F-&F+6$F4F0F1&F+6#-%\".G6$F4F4" } {TEXT -1 30 " is the sequence of entries " }{XPPEDIT 18 0 "a[1,1],a[ 2,2],` . . . `,a[n,n]" "6&&%\"aG6$\"\"\"F&&F$6$\"\"#F)%(~.~.~.~G&F$6$% \"nGF-" }{TEXT -1 35 " with the same row and column index" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 16 "The unit matrix " }{TEXT 304 1 "I" }{TEXT -1 136 " play s a role in relation to matrix multiplication analogous to that of the number 1 in relation to ordinary multiplication of numbers. " }} {PARA 0 "" 0 "" {TEXT -1 71 "One way of looking at division of numbers , is to say that the quotient " }{XPPEDIT 18 0 "a/b" "6#*&%\"aG\"\"\"% \"bG!\"\"" }{TEXT -1 16 " of two numbers " }{TEXT 293 1 "a" }{TEXT -1 5 " and " }{TEXT 294 1 "b" }{TEXT -1 19 " is the product of " }{TEXT 295 1 "a" }{TEXT -1 9 " and the " }{TEXT 259 22 "multiplicative invers e" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/b=b^(-1)" "6#/*&\"\"\"F%%\"bG!\" \")F&,$F%F'" }{TEXT -1 4 " of " }{TEXT 317 1 "b" }{TEXT -1 60 ". Such \+ a multiplicative inverse always exists provided that " }{XPPEDIT 18 0 "b<>0" "6#0%\"bG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "The multiplicative inverse " }{XPPEDIT 18 0 "1/b=b^(-1)" "6#/*&\"\"\" F%%\"bG!\"\")F&,$F%F'" }{TEXT -1 4 " of " }{TEXT 296 1 "b" }{TEXT -1 23 " has the property that " }{XPPEDIT 18 0 "b*b^(-1)=1" "6#/*&%\"bG\" \"\")F%,$F&!\"\"F&F&" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 38 "In a similar way the (multiplicative) " }{TEXT 259 7 "inverse" } {TEXT -1 20 " of a square matrix " }{TEXT 306 1 "A" }{TEXT -1 39 " (if it exists) is a matrix denoted by " }{XPPEDIT 18 0 "A^(-1)" "6#)%\"AG ,$\"\"\"!\"\"" }{TEXT -1 24 " with the property that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A*`.`*A^(-1)=I" "6#/*(%\"AG\"\"\"% \".GF&)F%,$F&!\"\"F&%\"IG" }{TEXT -1 9 " and " }{XPPEDIT 18 0 "A^( -1)*`.`*A=I" "6#/*()%\"AG,$\"\"\"!\"\"F(%\".GF(F&F(%\"IG" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 305 18 "_______________ ___" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 49 "There is a simple formula for the inverse of a 2 " } {TEXT 307 1 "x" }{TEXT -1 11 " 2 matrix " }{XPPEDIT 18 0 "A=matrix([[ a,b],[c,d]])" "6#/%\"AG-%'matrixG6#7$7$%\"aG%\"bG7$%\"cG%\"dG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "To find this formula first ob serve that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matr ix([[a,b],[c,d]])*matrix([[d,-b],[-c,a]])=matrix([[a*d-b*c,0],[0,a*d-b *c]])" "6#/*&-%'matrixG6#7$7$%\"aG%\"bG7$%\"cG%\"dG\"\"\"-F&6#7$7$F.,$ F+!\"\"7$,$F-F5F*F/-F&6#7$7$,&*&F*F/F.F/F/*&F+F/F-F/F5\"\"!7$F?,&*&F*F /F.F/F/*&F+F/F-F/F5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Hence, if " }{XPPEDIT 18 0 "a*d-b*c<>0 " "6#0,&*&%\"aG\"\"\"%\"dGF'F'*&%\"bGF'%\"cGF'!\"\"\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[a,b], [c,d]])*matrix([[d/(a*d-b*c),-b/(a*d-b*c)],[-c/(a*d-b*c),a/(a*d-b*c)]] )=matrix([[1,0],[0,1]])" "6#/*&-%'matrixG6#7$7$%\"aG%\"bG7$%\"cG%\"dG \"\"\"-F&6#7$7$*&F.F/,&*&F*F/F.F/F/*&F+F/F-F/!\"\"F8,$*&F+F/,&*&F*F/F. F/F/*&F+F/F-F/F8F8F87$,$*&F-F/,&*&F*F/F.F/F/*&F+F/F-F/F8F8F8*&F*F/,&*& F*F/F.F/F/*&F+F/F-F/F8F8F/-F&6#7$7$F/\"\"!7$FLF/" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "that is, \+ setting" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "A^(-1)=m atrix([[d/(a*d-b*c), -b/(a*d-b*c)], [-c/(a*d-b*c), a/(a*d-b*c)]])" "6# /)%\"AG,$\"\"\"!\"\"-%'matrixG6#7$7$*&%\"dGF',&*&%\"aGF'F/F'F'*&%\"bGF '%\"cGF'F(F(,$*&F4F',&*&F2F'F/F'F'*&F4F'F5F'F(F(F(7$,$*&F5F',&*&F2F'F/ F'F'*&F4F'F5F'F(F(F(*&F2F',&*&F2F'F/F'F'*&F4F'F5F'F(F(" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``=1/(a*d-b*c)" "6#/%!G*&\"\"\"F&,&*&%\"aGF&%\"dGF&F &*&%\"bGF&%\"cGF&!\"\"F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[d, \+ -b], [-c, a]]) " "6#-%'matrixG6#7$7$%\"dG,$%\"bG!\"\"7$,$%\"cGF+%\"aG " }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 308 32 "_ _______________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "we have " }{XPPEDIT 18 0 "A*`.`*A^(-1)=I" "6#/*(%\"AG\"\" \"%\".GF&)F%,$F&!\"\"F&%\"IG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "It is easy to see that " }{XPPEDIT 18 0 "A^(-1)*`.`*A=I \+ " "6#/*()%\"AG,$\"\"\"!\"\"F(%\".GF(F&F(%\"IG" }{TEXT -1 8 " also. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The qua ntity " }{XPPEDIT 18 0 "a*d-b*c" "6#,&*&%\"aG\"\"\"%\"dGF&F&*&%\"bGF&% \"cGF&!\"\"" }{TEXT -1 8 " is the " }{TEXT 259 11 "determinant" } {TEXT -1 15 " of the matrix " }{XPPEDIT 18 0 "matrix([[a,b],[c,d]])" " 6#-%'matrixG6#7$7$%\"aG%\"bG7$%\"cG%\"dG" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{GLPLOT2D 287 92 92 {PLOTDATA 2 "65-%'CUR VESG6$7&7$$!3))**************H!#=F(7$$!3++++++++]F*F(7$F,$\"3/+++++++8 !#<7$F(F/-%'COLOURG6&%$RGBG\"\"!F7F7-F$6$7&7$F/F(7$$\"3++++++++:F1F(7$ F=F/7$F/F/F3-F$6$7$7$$\"33+++++++EF1$!35+++++++?F*7$FE$\"3%*********** ***>\"F1F3-F$6$7$7$$\"3M+++++++WF1FG7$FPFJF3-%%TEXTG6&7$$F7F7$\"\"\"F7 Q\"a6\"F3-%%FONTG6%%&TIMESG%'ITALICG\"#7-FT6&7$FXFXQ\"bFenF3Ffn-FT6&7$ FWFWQ\"cFenF3Ffn-FT6&7$FXFWQ\"dFenF3Ffn-FT6&7$$\"\"$F7FXFZF3Ffn-FT6&7$ $\"\"%F7FXF_oF3Ffn-FT6&7$F[pFWFcoF3Ffn-FT6&7$F`pFWFgoF3Ffn-FT6&7$$\"\" #F7$\"\"&!\"\"Q\"=FenF3Ffn-FT6&7$$\"#]F_qF]qF`qF3Ffn-FT6&7$$\"#jF_qF]q Q*a~d~-~b~cFenF3Ffn-FT6&7$$!\"*F_qF]qQ$detFenF3-Fgn6%Fin%&ROMANGF[o-%* AXESSTYLEG6#%%NONEG-%+AXESLABELSG6$Q!FenF\\s-%%VIEWG6$%(DEFAULTGF`s" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Cur ve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15 " "Curve 16" }}{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 28 "Thus we \+ see that a square 2 " }{TEXT 309 1 "x" }{TEXT -1 97 " 2 matrix A has i n inverse exactly when its determinant is non-zero. Such a matrix is s aid to be " }{TEXT 259 10 "invertible" }{TEXT -1 10 " or to be " } {TEXT 259 12 "non-singular" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 310 8 "Question" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 7 "Given " }{XPPEDIT 18 0 "A=matrix([[2,5],[ 1,3]])" "6#/%\"AG-%'matrixG6#7$7$\"\"#\"\"&7$\"\"\"\"\"$" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "B=matrix([[1,-1],[2,3]])" "6#/%\"BG-%'matrixG 6#7$7$\"\"\",$F*!\"\"7$\"\"#\"\"$" }{TEXT -1 8 ", find " }{XPPEDIT 18 0 "A^(-1)" "6#)%\"AG,$\"\"\"!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "B^(-1)" "6#)%\"BG,$\"\"\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "A*B " "6#*&%\"AG\"\"\"%\"BGF%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "(A*B)^(-1 )" "6#)*&%\"AG\"\"\"%\"BGF&,$F&!\"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "B^(-1)*A^(-1)" "6#*&)%\"BG,$\"\"\"!\"\"F')%\"AG,$F'F(F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 311 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 38 "Using the formual given above, since " }{XPPEDIT 18 0 "det(A) = ``(2)*``(3)-`` (1)*``(5);" "6#/-%$detG6#%\"AG,&*&-%!G6#\"\"#\"\"\"-F+6#\"\"$F.F.*&-F+ 6#F.F.-F+6#\"\"&F.!\"\"" }{XPPEDIT 18 0 "``=1" "6#/%!G\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "A^(-1)=matrix([[3,-5],[-1,2]])" "6#/)%\"AG, $\"\"\"!\"\"-%'matrixG6#7$7$\"\"$,$\"\"&F(7$,$F'F(\"\"#" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 12 "and, since " }{XPPEDIT 18 0 "det( B)=``(1)*``(3)-``(2)*``(-1)" "6#/-%$detG6#%\"BG,&*&-%!G6#\"\"\"F--F+6# \"\"$F-F-*&-F+6#\"\"#F--F+6#,$F-!\"\"F-F8" }{XPPEDIT 18 0 "``=5" "6#/% !G\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "B^(-1)=1/5" "6#/)%\"BG,$\" \"\"!\"\"*&F'F'\"\"&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[3,1], [-2,1]])=matrix([[3/5,1/5],[-2/5,1/5]])" "6#/-%'matrixG6#7$7$\"\"$\"\" \"7$,$\"\"#!\"\"F*-F%6#7$7$*&F)F*\"\"&F.*&F*F*F4F.7$,$*&F-F*F4F.F.*&F* F*F4F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " } {XPPEDIT 18 0 "A*B=matrix([[2,5],[1,3]])*matrix([[1,-1],[2,3]])" "6#/* &%\"AG\"\"\"%\"BGF&*&-%'matrixG6#7$7$\"\"#\"\"&7$F&\"\"$F&-F*6#7$7$F&, $F&!\"\"7$F.F1F&" }{XPPEDIT 18 0 "``=matrix([[12,13],[7,8]])" "6#/%!G- %'matrixG6#7$7$\"#7\"#87$\"\"(\"\")" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 6 "Since " }{XPPEDIT 18 0 "det(A*B)=``(12)*``(8)-``(7)*``(1 3)" "6#/-%$detG6#*&%\"AG\"\"\"%\"BGF),&*&-%!G6#\"#7F)-F.6#\"\")F)F)*&- F.6#\"\"(F)-F.6#\"#8F)!\"\"" }{XPPEDIT 18 0 "``=5" "6#/%!G\"\"&" } {TEXT -1 4 ", " }{XPPEDIT 18 0 "(A*B)^(-1)=1/5" "6#/)*&%\"AG\"\"\"% \"BGF',$F'!\"\"*&F'F'\"\"&F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([ [8,-13],[-7,12]])=matrix([[8/5,-13/5],[-7/5,12/5]])" "6#/-%'matrixG6#7 $7$\"\"),$\"#8!\"\"7$,$\"\"(F,\"#7-F%6#7$7$*&F)\"\"\"\"\"&F,,$*&F+F6F7 F,F,7$,$*&F/F6F7F,F,*&F0F6F7F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 9 "Finally, " }{XPPEDIT 18 0 "B^(-1)*A^(-1)=matrix([[3/5, 1/5 ], [-2/5, 1/5]])*matrix([[3, -5], [-1, 2]])" "6#/*&)%\"BG,$\"\"\"!\"\" F()%\"AG,$F(F)F(*&-%'matrixG6#7$7$*&\"\"$F(\"\"&F)*&F(F(F5F)7$,$*&\"\" #F(F5F)F)*&F(F(F5F)F(-F/6#7$7$F4,$F5F)7$,$F(F)F:F(" }{XPPEDIT 18 0 "`` =matrix([[8/5,-13/5],[-7/5,12/5]])" "6#/%!G-%'matrixG6#7$7$*&\"\")\"\" \"\"\"&!\"\",$*&\"#8F,F-F.F.7$,$*&\"\"(F,F-F.F.*&\"#7F,F-F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "(A*B)^(-1)=B^(-1)*A^(-1)" "6#/)*&%\"AG \"\"\"%\"BGF',$F'!\"\"*&)F(,$F'F*F')F&,$F'F*F'" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 31 "This result is true in general." }}{PARA 0 "" 0 "" {TEXT -1 13 "To show that " }{XPPEDIT 18 0 "U=B^(-1)*A^(-1) " "6#/%\"UG*&)%\"BG,$\"\"\"!\"\"F))%\"AG,$F)F*F)" }{TEXT -1 19 " is th e inverse of " }{XPPEDIT 18 0 "V=A*B" "6#/%\"VG*&%\"AG\"\"\"%\"BGF'" } {TEXT -1 23 ", we need to show that " }{XPPEDIT 18 0 "U*V=I" "6#/*&%\" UG\"\"\"%\"VGF&%\"IG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "V*U=I" "6# /*&%\"VG\"\"\"%\"UGF&%\"IG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 "Now" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "U*V=``( B^(-1)*A^(-1))*``(A*B)" "6#/*&%\"UG\"\"\"%\"VGF&*&-%!G6#*&)%\"BG,$F&! \"\"F&)%\"AG,$F&F0F&F&-F*6#*&F2F&F.F&F&" }{XPPEDIT 18 0 " ``= B^(-1)*A ^(-1)*A*B" "6#/%!G**)%\"BG,$\"\"\"!\"\"F))%\"AG,$F)F*F)F,F)F'F)" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 86 " (Brackets are not nece ssary because of the associativity of matrix multiplication.) " }} {PARA 0 "" 0 "" {TEXT -1 7 " Hence " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "U*V=B^(-1)*I*B" "6#/*&%\"UG\"\"\"%\"VGF&*()%\"BG,$F &!\"\"F&%\"IGF&F*F&" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "B^(-1)*B=I" "6# /*&)%\"BG,$\"\"\"!\"\"F(F&F(%\"IG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "One can show that " }{XPPEDIT 18 0 "V*U=I" "6#/*&%\"VG\" \"\"%\"UGF&%\"IG" }{TEXT -1 19 " in a similar way. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 53 "Using matrices to solve a system \+ of linear equations " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 40 "Consider the system of linear equations \+ " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([a*x+b *y = p, ``],[c*x+d*y = q, ``]);" "6#-%*PIECEWISEG6$7$/,&*&%\"aG\"\"\"% \"xGF+F+*&%\"bGF+%\"yGF+F+%\"pG%!G7$/,&*&%\"cGF+F,F+F+*&%\"dGF+F/F+F+% \"qGF1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "This system ca n be written in the matrix form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[a,b],[c,d]])*matrix([[x],[y]])=matrix([[p],[q ]])" "6#/*&-%'matrixG6#7$7$%\"aG%\"bG7$%\"cG%\"dG\"\"\"-F&6#7$7#%\"xG7 #%\"yGF/-F&6#7$7#%\"pG7#%\"qG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A*`.`*X=P" "6#/*(%\"AG\"\"\"%\".GF&%\"XGF&%\"PG" }{TEXT -1 15 " \+ ------- (i), " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 " A=matrix([[a, b], [c, d]])" "6#/%\"AG-%'matrixG6#7$7$%\"aG%\"bG7$%\"cG %\"dG" }{TEXT -1 9 " is the " }{TEXT 259 18 "coefficient matrix" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "P=matrix([[p],[q]])" "6#/%\"PG-%'matri xG6#7$7#%\"pG7#%\"qG" }{TEXT -1 109 " is a column vector whose entrie s are the constants obtained from the right sides of the two equations , and " }{XPPEDIT 18 0 "X=matrix([[x],[y]])" "6#/%\"XG-%'matrixG6#7$7# %\"xG7#%\"yG" }{TEXT -1 84 " is a column vector whose entries are the \+ \"unknowns\" in the two equations, that is, " }{TEXT 316 1 "X" }{TEXT -1 8 " is the " }{TEXT 259 15 "solution vector" }{TEXT -1 17 " for the system. " }}{PARA 0 "" 0 "" {TEXT -1 90 "Suppose that the coefficient matrix A is non-singular (invertible), that is, suppose that " } {XPPEDIT 18 0 "det(A)<>0" "6#0-%$detG6#%\"AG\"\"!" }{TEXT -1 26 ". The n the inverse matrix " }{XPPEDIT 18 0 "A^(-1)" "6#)%\"AG,$\"\"\"!\"\" " }{TEXT -1 15 " can be found. " }}{PARA 0 "" 0 "" {TEXT -1 77 "The eq uation (i) can then be solved by multiplying both sides on the left by " }{XPPEDIT 18 0 "A^(-1)" "6#)%\"AG,$\"\"\"!\"\"" }{TEXT -1 10 " to g ive: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A^(-1)*`.`*A *`.`*X=A^(-1)*`.`*P" "6#/*,)%\"AG,$\"\"\"!\"\"F(%\".GF(F&F(F*F(%\"XGF( *()F&,$F(F)F(F*F(%\"PGF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I*`.`* X = A^(-1)*`.`*P;" "6#/*(%\"IG\"\"\"%\".GF&%\"XGF&*()%\"AG,$F&!\"\"F&F 'F&%\"PGF&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "X=A^(-1)*`.`*P" "6#/ %\"XG*()%\"AG,$\"\"\"!\"\"F)%\".GF)%\"PGF)" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 312 7 "_______" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 313 8 "Question " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 35 "Solve the system of \+ equations (a) " }{XPPEDIT 18 0 "PIECEWISE([3*x+2*y=4,``],[x-y=3,``]) " "6#-%*PIECEWISEG6$7$/,&*&\"\"$\"\"\"%\"xGF+F+*&\"\"#F+%\"yGF+F+\"\"% %!G7$/,&F,F+F/!\"\"F*F1" }{TEXT -1 9 "and (b) " }{XPPEDIT 18 0 "PIECE WISE([3*x+2*y = 5, ``],[x-y = 4, ``])" "6#-%*PIECEWISEG6$7$/,&*&\"\"$ \"\"\"%\"xGF+F+*&\"\"#F+%\"yGF+F+\"\"&%!G7$/,&F,F+F/!\"\"\"\"%F1" } {TEXT -1 49 " by using the inverse of the coefficient matrix. " }} {PARA 0 "" 0 "" {TEXT 314 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 48 "(a) This system of equations in matrix form is: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[3,2], [1,-1] ])*matrix([[x], [y]]) = matrix([[4], [3]])" "6#/*&-%'matrixG6#7$7$\"\" $\"\"#7$\"\"\",$F-!\"\"F--F&6#7$7#%\"xG7#%\"yGF--F&6#7$7#\"\"%7#F*" } {TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 3 "or " }{TEXT 315 0 "" } }{PARA 256 "" 0 "" {XPPEDIT 18 0 "A*`.`*X=P" "6#/*(%\"AG\"\"\"%\".GF&% \"XGF&%\"PG" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "A=matrix([[3,2], [1,-1]])" "6#/%\"AG-%'matrixG6#7$7$\" \"$\"\"#7$\"\"\",$F-!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "P=matrix( [[4], [3]])" "6#/%\"PG-%'matrixG6#7$7#\"\"%7#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "X=matrix([[x], [y]])" "6#/%\"XG-%'matrixG6#7$7#%\" xG7#%\"yG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 41 "The inverse of the coefficient matrix is " }{XPPEDIT 18 0 "A^(-1) = 1/(-5);" "6#/ )%\"AG,$\"\"\"!\"\"*&F'F',$\"\"&F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 " matrix([[-1,-2],[-1,3]])=matrix([[1/5,2/5],[1/5,-3/5]])" "6#/-%'matrix G6#7$7$,$\"\"\"!\"\",$\"\"#F+7$,$F*F+\"\"$-F%6#7$7$*&F*F*\"\"&F+*&F-F* F6F+7$*&F*F*F6F+,$*&F0F*F6F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "The solution vector is " }{XPPEDIT 18 0 "X=A^(-1)*`.`*P" "6#/%\"XG*()%\"AG,$\"\"\"!\"\"F)%\".GF )%\"PGF)" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "matrix([[1/5,2/5],[1/5,-3/ 5]])*matrix([[4], [3]])=matrix([[4/5+6/5],[4/5-9/5]])" "6#/*&-%'matrix G6#7$7$*&\"\"\"F+\"\"&!\"\"*&\"\"#F+F,F-7$*&F+F+F,F-,$*&\"\"$F+F,F-F-F +-F&6#7$7#\"\"%7#F4F+-F&6#7$7#,&*&F9F+F,F-F+*&\"\"'F+F,F-F+7#,&*&F9F+F ,F-F+*&\"\"*F+F,F-F-" }{XPPEDIT 18 0 "``=matrix([[2],[-1]])" "6#/%!G-% 'matrixG6#7$7#\"\"#7#,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 40 "The solution of the system is therefore " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y=-1" "6#/%\" yG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 51 "It is easy to check that these values are correct. " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 141 ": For a single system of equations t his matrix method involves more work than a traditional method based o n eliminating one of the variables." }}{PARA 0 "" 0 "" {TEXT -1 196 "H owever, an advantage is that, a second system with the same coefficien t matrix, but with a different vector of constants (as in part(b) ) ca n readily be solved by a single matrix multiplication. " }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{PARA 0 "" 0 "" {TEXT -1 5 " Let " }{XPPEDIT 18 0 "Q = matrix([[5], [4]])" "6#/%\"QG-%'matrixG6#7$7#\"\"&7#\"\"%" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "Then the solution of " }{XPPEDIT 18 0 "A*`.`*X = Q" "6#/*(%\"AG\"\"\"%\".GF&%\"XGF&%\"QG" } {TEXT -1 5 " is " }{XPPEDIT 18 0 "X = A^(-1)*`.`*Q" "6#/%\"XG*()%\"AG ,$\"\"\"!\"\"F)%\".GF)%\"QGF)" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "matri x([[1/5, 2/5], [1/5, -3/5]])*matrix([[5], [4]]) = matrix([[1+8/5], [1- 12/5]])" "6#/*&-%'matrixG6#7$7$*&\"\"\"F+\"\"&!\"\"*&\"\"#F+F,F-7$*&F+ F+F,F-,$*&\"\"$F+F,F-F-F+-F&6#7$7#F,7#\"\"%F+-F&6#7$7#,&F+F+*&\"\")F+F ,F-F+7#,&F+F+*&\"#7F+F,F-F-" }{XPPEDIT 18 0 "`` = matrix([[13/5], [-7/ 5]])" "6#/%!G-%'matrixG6#7$7#*&\"#8\"\"\"\"\"&!\"\"7#,$*&\"\"(F,F-F.F. " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 12 "which gives " } {XPPEDIT 18 0 "x=13/5" "6#/%\"xG*&\"#8\"\"\"\"\"&!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y=-7/5" "6#/%\"yG,$*&\"\"(\"\"\"\"\"&!\"\"F*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 60 ": T he solution can be checked by the matrix multiplication: " }{XPPEDIT 18 0 "matrix([[3, 2], [1, -1]])*matrix([[13/5], [-7/5]])=matrix([[5], \+ [4]])" "6#/*&-%'matrixG6#7$7$\"\"$\"\"#7$\"\"\",$F-!\"\"F--F&6#7$7#*& \"#8F-\"\"&F/7#,$*&\"\"(F-F6F/F/F--F&6#7$7#F67#\"\"%" }{TEXT -1 2 ". \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Perform the following matrix multiplications: " }}{PARA 0 "" 0 "" {TEXT -1 12 " (a) " }{XPPEDIT 18 0 "matrix([[0, -1, 2], [4, 11, 2]])*matrix([[3, -1], [1, 2], [6, 1]]);" "6#*&-%'matrixG6#7$7%\"\"!,$ \"\"\"!\"\"\"\"#7%\"\"%\"#6F-F+-F%6#7%7$\"\"$,$F+F,7$F+F-7$\"\"'F+F+" }{TEXT -1 24 " (b) " }{XPPEDIT 18 0 "matrix([[-1, 7 ], [3, 5], [10, -1], [-5, 12]])*matrix([[2, 1], [5, -3]]);" "6#*&-%'ma trixG6#7&7$,$\"\"\"!\"\"\"\"(7$\"\"$\"\"&7$\"#5,$F*F+7$,$F/F+\"#7F*-F% 6#7$7$\"\"#F*7$F/,$F.F+F*" }{TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 12 " (c) " }{XPPEDIT 18 0 "matrix([[-7, 8], [5, 2]])*m atrix([[-6, 10], [1, 4]]);" "6#*&-%'matrixG6#7$7$,$\"\"(!\"\"\"\")7$\" \"&\"\"#\"\"\"-F%6#7$7$,$\"\"'F+\"#57$F0\"\"%F0" }{TEXT -1 33 " \+ (d) " }{XPPEDIT 18 0 "matrix([[1, 2, -6, 6, 1], \+ [-2, 4, 0, 1, 2]])*matrix([[1], [-1], [0], [5], [2]]);" "6#*&-%'matrix G6#7$7'\"\"\"\"\"#,$\"\"'!\"\"F,F)7',$F*F-\"\"%\"\"!F)F*F)-F%6#7'7#F)7 #,$F)F-7#F17#\"\"&7#F*F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "__ _________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }} {PARA 0 "" 0 "" {TEXT -1 39 "Find, if possible, the matrix products " }{XPPEDIT 18 0 "A*B" "6#*&%\"AG\"\"\"%\"BGF%" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "B*A" "6#*&%\"BG\"\"\"%\"AGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 " (a) " } {XPPEDIT 18 0 "A=matrix([[1,-3,8]])" "6#/%\"AG-%'matrixG6#7#7%\"\"\",$ \"\"$!\"\"\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "B = matrix([[-1], [ 5], [7]]);" "6#/%\"BG-%'matrixG6#7%7#,$\"\"\"!\"\"7#\"\"&7#\"\"(" } {TEXT -1 14 " (b) " }{XPPEDIT 18 0 "A=matrix([[-1,2,3],[5,-1, 0]])" "6#/%\"AG-%'matrixG6#7$7%,$\"\"\"!\"\"\"\"#\"\"$7%\"\"&,$F+F,\" \"!" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "B=matrix([[1],[-5],[2]])" "6#/ %\"BG-%'matrixG6#7%7#\"\"\"7#,$\"\"&!\"\"7#\"\"#" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 24 "Find the inverse matri x " }{XPPEDIT 18 0 "A^(-1)" "6#)%\"AG,$\"\"\"!\"\"" }{TEXT -1 49 " in \+ each of the following cases. Also check that " }{XPPEDIT 18 0 "A^(-1)* `.`*A=I" "6#/*()%\"AG,$\"\"\"!\"\"F(%\".GF(F&F(%\"IG" }{TEXT -1 5 " an d " }{XPPEDIT 18 0 "A*`.`*A^(-1)=I" "6#/*(%\"AG\"\"\"%\".GF&)F%,$F&!\" \"F&%\"IG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "A = matrix([[1, -5], [1, \+ -4]]);" "6#/%\"AG-%'matrixG6#7$7$\"\"\",$\"\"&!\"\"7$F*,$\"\"%F-" } {TEXT -1 10 " (b) " }{XPPEDIT 18 0 "A = matrix([[2, -1], [4, 6]]) ;" "6#/%\"AG-%'matrixG6#7$7$\"\"#,$\"\"\"!\"\"7$\"\"%\"\"'" }{TEXT -1 10 " (c) " }{XPPEDIT 18 0 "A = matrix([[-4, 3], [-3, 5]]);" "6#/% \"AG-%'matrixG6#7$7$,$\"\"%!\"\"\"\"$7$,$F-F,\"\"&" }{TEXT -1 2 " " } }{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "A = matrix([[2, 1], [5, 3]]);" "6#/%\"AG-%'matrixG6#7$7$\"\"#\"\"\"7$\" \"&\"\"$" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "B = matrix([[1, -1], [4 , 6]]);" "6#/%\"BG-%'matrixG6#7$7$\"\"\",$F*!\"\"7$\"\"%\"\"'" }{TEXT -1 9 ", find: " }}{PARA 0 "" 0 "" {TEXT -1 5 " (a) " }{XPPEDIT 18 0 " 2*A-3*B" "6#,&*&\"\"#\"\"\"%\"AGF&F&*&\"\"$F&%\"BGF&!\"\"" }{TEXT -1 10 " (b) " }{XPPEDIT 18 0 "A*B" "6#*&%\"AG\"\"\"%\"BGF%" }{TEXT -1 10 " (c) " }{XPPEDIT 18 0 "A^(-1)" "6#)%\"AG,$\"\"\"!\"\"" } {TEXT -1 9 " (d) " }{XPPEDIT 18 0 "B^(-1)" "6#)%\"BG,$\"\"\"!\"\" " }{TEXT -1 8 " (e) " }{XPPEDIT 18 0 "(A*B)^(-1)" "6#)*&%\"AG\"\"\" %\"BGF&,$F&!\"\"" }{TEXT -1 9 " (f) " }{XPPEDIT 18 0 "B^(-1)*A^(-1 )" "6#*&)%\"BG,$\"\"\"!\"\"F')%\"AG,$F'F(F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "Solv e the following systems of equations by using the inverse of the coeff icient matrix. " }}{PARA 0 "" 0 "" {TEXT -1 12 " (a) " } {XPPEDIT 18 0 "PIECEWISE([2*x-5*y=-14,``],[-2*x+4*y=11,``])" "6#-%*PIE CEWISEG6$7$/,&*&\"\"#\"\"\"%\"xGF+F+*&\"\"&F+%\"yGF+!\"\",$\"#9F0%!G7$ /,&*&F*F+F,F+F0*&\"\"%F+F/F+F+\"#6F3" }{TEXT -1 9 " (b) " } {XPPEDIT 18 0 "PIECEWISE([5*x-2*y=15,``],[3*x+4*y=7,``])" "6#-%*PIECEW ISEG6$7$/,&*&\"\"&\"\"\"%\"xGF+F+*&\"\"#F+%\"yGF+!\"\"\"#:%!G7$/,&*&\" \"$F+F,F+F+*&\"\"%F+F/F+F+\"\"(F2" }{TEXT -1 7 " (c) " }{XPPEDIT 18 0 "PIECEWISE([5*x-2*y=10,``],[3*x+4*y=1,``])" "6#-%*PIECEWISEG6$7$/,&* &\"\"&\"\"\"%\"xGF+F+*&\"\"#F+%\"yGF+!\"\"\"#5%!G7$/,&*&\"\"$F+F,F+F+* &\"\"%F+F/F+F+F+F2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 17 "C ode for pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 0 "" 0 "" {TEXT -1 28 "multiplication of matrices 1" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2109 "p1:=plot([[[.2,0],[0,0],[0,6],[.2,6]],\n [[9.8,0],[10,0],[10,6 ],[9.8,6]],\n [[11.2,-3],[11,-3],[11,6],[11.2,6]],\n [[16.8,-3],[1 7,-3],[17,6],[16.8,6]],\n [[20.2,0],[20,0],[20,6],[20.2,6]],\n [[2 5.8,0],[26,0],[26,6],[25.8,6]]],color=black):\np2 := plottools[arrow]( [-2,2.5],[-2,0],0,.3,.12,arrow,color=blue):\np3 := plottools[arrow]([- 2,3.5],[-2,6],0,.3,.12,arrow,color=blue):\np4 := plottools[arrow]([3,7 ],[0,7],0,.3,.12,arrow,color=blue):\np5 := plottools[arrow]([7,7],[10, 7],0,.3,.12,arrow,color=blue):\np6 := plottools[arrow]([12.5,7],[11,7] ,0,.3,.25,arrow,color=COLOR(RGB,0,.7,0)):\np7 := plottools[arrow]([15. 5,7],[17,7],0,.3,.25,arrow,color=COLOR(RGB,0,.7,0)):\np8 := plottools[ arrow]([12.5,1.5],[12.5,6],0,.3,.08,arrow,color=COLOR(RGB,0,.7,0)):\np 9 := plottools[arrow]([12.5,.5],[12.5,-3],0,.3,.08,arrow,color=COLOR(R GB,0,.7,0)):\np10 := plottools[arrow]([28,2.5],[28,0],0,.3,.12,arrow,c olor=COLOR(RGB,.8,0,.8)):\np11 := plottools[arrow]([28,3.5],[28,6],0,. 3,.12,arrow,color=COLOR(RGB,.8,0,.8)):\np12 := plottools[arrow]([21.5, 7],[20,7],0,.3,.25,arrow,color=COLOR(RGB,.8,0,.8)):\np13 := plottools[ arrow]([24.5,7],[26,7],0,.3,.25,arrow,color=COLOR(RGB,.8,0,.8)):\np14 \+ := plot([[[.3,4],[9.7,4]],[[14.3,-3],[14.3,6]]],color=red,thickness=3) :\np15 := plot([[[20.3,4],[25.7,4]],[[23.3,0],[23.3,6]]],color=red,lin estyle=2):\np16 := plot([[[23.3,4]]$3],style=point,color=red,symbol=[c ircle,diamond,cross]):\np17 := plots[polygonplot]([[[0,0],[0,6],[10,6] ,[10,0]],\n [[11,-3],[11,6],[17,6],[17,-3]],[[20,0],[20,6],[26,6],[26 ,0]]],\n style=patchnogrid,color=COLOR(RGB,.95,.95,.95) ):\nt1 := plots[textplot]([18.5,2.9,`=`],color=black):\nt2 := plots[te xtplot]([[-2,3,`m rows`],[5,7,`n cols`]],color=blue):\nt3 := plots[tex tplot]([[12.5,1,`n rows`],[14,7,`p cols`]],color=COLOR(RGB,0,.7,0)):\n t4 := plots[textplot]([[5,3.5,`i th row`],[15.8,2.6,`j th col`],\n \+ [20.65,3.6,`i`],[23.6,5.3,`j`]],color=red):\nt5 := plots[textplot]([ [28,3,`m rows`],[23,7,`p cols`]],color=COLOR(RGB,.8,0,.8)):\nplots[dis play]([p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12,p13,\n \+ p14,p15,p16,p17,t1,t2,t3,t4,t5],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 28 "multiplication of matrices 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1168 " p1 := plot([[[-.3,-.3],[-.5,-.3],[-.5,2.3],[-.3,2.3]],\n [[3.3,-.3], [3.5,-.3],[3.5,2.3],[3.3,2.3]],\n [[4.2,-1.3],[4,-1.3],[4,2.3],[4.2, 2.3]],\n [[8.8,-1.3],[9,-1.3],[9,2.3],[8.8,2.3]],\n [[10.7,-.3],[10. 5,-.3],[10.5,2.3],[10.7,2.3]],\n [[15.3,-.3],[15.5,-.3],[15.5,2.3],[1 5.3,2.3]]],color=black):\np2 := plot([[[-.3,1.8],[-.3,2.2],[3.2,2.2],[ 3.2,1.8],[-.3,1.8]],\n [[4.2,2.2],[4.7,2.2],[4.7,-1.2],[4.2,-1.2],[4 .2,2.2]]],color=red):\nd := evalf(Pi/10):\np3 := plot([seq([11+.33*cos (i*d),2+.33*sin(i*d)],i=0..20)]): \nt1 := plots[textplot]([[0,0,`-4`], [1,0,`6`],[2,0,`-2`],\n[3,0,`1`],[0,1,`2`],[1,1,`-1`],[2,1,`4`],[3,1,` 3`],\n[0,2,`3`],[1,2,`5`],[2,2,`1`],[3,2,`2`],\n[4.5,2,`2`],[5.5,2,`1` ],[6.5,2,`4`],[7.5,2,`3`],[8.5,2,`-2`],\n[4.5,1,`5`],[5.5,1,`2`],[6.5, 1,`-1`],[7.5,1,`7`],[8.5,1,`1`],\n[4.5,0,`-2`],[5.5,0,`6`],[6.5,0,`3`] ,[7.5,0,`4`],[8.5,0,`5`],\n[4.5,-1,`3`],[5.5,-1,`-2`],[6.5,-1,`5`],[7. 5,-1,`1`],[8.5,-1,`2`],\n[11,2,`35`],[12,2,`15`],[13,2,`20`],[14,2,`50 `],[15,2,`8`],\n[11,1,`0`],[12,1,`18`],[13,1,`36`],[14,1,`18`],[15,1,` 21`],\n[11,0,`29`],[12,0,`-6`],[13,0,`-23`],[14,0,`23`],[15,0,`6`],\n \+ [9.75,1,`=`]],color=black):\nplots[display]([p1,p2,p3,t1],axes=none); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "A := Matrix([[3,5,1,2],[2,- 1,4,3],[-4,6,-2,1]]):\nB := Matrix([[2,1,4,3,-2],[5,2,-1,7,1],[-2,6,3, 4,5],[3,-2,5,1,2]]):\nA,`.`,B=A.B;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1168 "p1 := plot([[[-.3,-.3],[- .5,-.3],[-.5,2.3],[-.3,2.3]],\n [[3.3,-.3],[3.5,-.3],[3.5,2.3],[3.3, 2.3]],\n [[4.2,-1.3],[4,-1.3],[4,2.3],[4.2,2.3]],\n [[8.8,-1.3],[9, -1.3],[9,2.3],[8.8,2.3]],\n [[10.7,-.3],[10.5,-.3],[10.5,2.3],[10.7,2 .3]],\n [[15.3,-.3],[15.5,-.3],[15.5,2.3],[15.3,2.3]]],color=black): \np2 := plot([[[-.3,1.8],[-.3,2.2],[3.2,2.2],[3.2,1.8],[-.3,1.8]],\n \+ [[5.2,2.2],[5.7,2.2],[5.7,-1.2],[5.2,-1.2],[5.2,2.2]]],color=red):\nd := evalf(Pi/10):\np3 := plot([seq([12+.33*cos(i*d),2+.33*sin(i*d)],i= 0..20)]): \nt1 := plots[textplot]([[0,0,`-4`],[1,0,`6`],[2,0,`-2`],\n[ 3,0,`1`],[0,1,`2`],[1,1,`-1`],[2,1,`4`],[3,1,`3`],\n[0,2,`3`],[1,2,`5` ],[2,2,`1`],[3,2,`2`],\n[4.5,2,`2`],[5.5,2,`1`],[6.5,2,`4`],[7.5,2,`3` ],[8.5,2,`-2`],\n[4.5,1,`5`],[5.5,1,`2`],[6.5,1,`-1`],[7.5,1,`7`],[8.5 ,1,`1`],\n[4.5,0,`-2`],[5.5,0,`6`],[6.5,0,`3`],[7.5,0,`4`],[8.5,0,`5`] ,\n[4.5,-1,`3`],[5.5,-1,`-2`],[6.5,-1,`5`],[7.5,-1,`1`],[8.5,-1,`2`], \n[11,2,`35`],[12,2,`15`],[13,2,`20`],[14,2,`50`],[15,2,`8`],\n[11,1,` 0`],[12,1,`18`],[13,1,`36`],[14,1,`18`],[15,1,`21`],\n[11,0,`29`],[12, 0,`-6`],[13,0,`-23`],[14,0,`23`],[15,0,`6`],\n [9.75,1,`=`]],color=bla ck):\nplots[display]([p1,p2,p3,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1165 "p1 := plot ([[[-.3,-.3],[-.5,-.3],[-.5,2.3],[-.3,2.3]],\n [[3.3,-.3],[3.5,-.3], [3.5,2.3],[3.3,2.3]],\n [[4.2,-1.3],[4,-1.3],[4,2.3],[4.2,2.3]],\n \+ [[8.8,-1.3],[9,-1.3],[9,2.3],[8.8,2.3]],\n [[10.7,-.3],[10.5,-.3],[10 .5,2.3],[10.7,2.3]],\n [[15.3,-.3],[15.5,-.3],[15.5,2.3],[15.3,2.3]]] ,color=black):\np2 := plot([[[-.3,.8],[-.3,1.2],[3.2,1.2],[3.2,.8],[-. 3,.8]],\n [[7.2,2.2],[7.7,2.2],[7.7,-1.2],[7.2,-1.2],[7.2,2.2]]],col or=red):\nd := evalf(Pi/10):\np3 := plot([seq([14+.33*cos(i*d),1+.33*s in(i*d)],i=0..20)]): \nt1 := plots[textplot]([[0,0,`-4`],[1,0,`6`],[2, 0,`-2`],\n[3,0,`1`],[0,1,`2`],[1,1,`-1`],[2,1,`4`],[3,1,`3`],\n[0,2,`3 `],[1,2,`5`],[2,2,`1`],[3,2,`2`],\n[4.5,2,`2`],[5.5,2,`1`],[6.5,2,`4`] ,[7.5,2,`3`],[8.5,2,`-2`],\n[4.5,1,`5`],[5.5,1,`2`],[6.5,1,`-1`],[7.5, 1,`7`],[8.5,1,`1`],\n[4.5,0,`-2`],[5.5,0,`6`],[6.5,0,`3`],[7.5,0,`4`], [8.5,0,`5`],\n[4.5,-1,`3`],[5.5,-1,`-2`],[6.5,-1,`5`],[7.5,-1,`1`],[8. 5,-1,`2`],\n[11,2,`35`],[12,2,`15`],[13,2,`20`],[14,2,`50`],[15,2,`8`] ,\n[11,1,`0`],[12,1,`18`],[13,1,`36`],[14,1,`18`],[15,1,`21`],\n[11,0, `29`],[12,0,`-6`],[13,0,`-23`],[14,0,`23`],[15,0,`6`],\n [9.75,1,`=`]] ,color=black):\nplots[display]([p1,p2,p3,t1],axes=none);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 0 "" 0 "" {TEXT -1 12 "determinant " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 461 "p1 := plot( [[[-.3,-.3],[-.5,-.3],[-.5,1.3],[-.3,1.3]],\n [[1.3,-.3],[1.5,-.3],[1 .5,1.3],[1.3,1.3]],\n [[2.6,-.2],[2.6,1.2]],[[4.4,-.2],[4.4,1.2]]],co lor=black):\nt1 := plots[textplot]([[0,1,`a`],[1,1,`b`],[0,0,`c`],[1,0 ,`d`],\n [3,1,`a`],[4,1,`b`],[3,0,`c`],[4,0,`d`],\n [2,.5,`=`],[5.0,.5 ,`=`],[6.3,.5,`a d - b c`]],font=[TIMES,ITALIC,12],color=black):\nt2 : = plots[textplot]([-.9,.5,`det`],font=[TIMES,ROMAN,12],color=black):\n plots[display]([p1,t1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 145461596 145463528 145465168 145467940 145471108 145473808 145477876 145480648 145482052 145484752 145486444 145489216 145490188 145493032 145494580 145498972 145500448 145505740 145507216 145509916 145511320 145514020 145515424 145518196 145521292 145524280 145528348 145532992 145534468 145537312 }{RTABLE M7R0 I6RTABLE_SAVE/145461596X,%)anythingG6"6"[gl!"%!!!#-"$"%""$""#!"%""&!""""'"""""% !"#F(F'F-6" } {RTABLE M7R0 I6RTABLE_SAVE/145463528X,%)anythingG6"6"[gl!"%!!!#5"%"&""#""&!"#""$"""F'""'F)"" %!""F*F(F*""(F-F+F)F+F(F'6" } {RTABLE M7R0 I6RTABLE_SAVE/145465168X,%)anythingG6"6"[gl!"%!!!#0"$"&"#N""!"#H"#:"#=!"'"#?"#O !#B"#]F+"#B"")"#@""'6" } {RTABLE M7R0 I6RTABLE_SAVE/145467940X,%)anythingG6"6"[gl!"%!!!#%"#"#"""""$""#""%6" } {RTABLE M7R0 I6RTABLE_SAVE/145471108X,%)anythingG6"6"[gl!"%!!!#%"#"#!"#""&"""""$6" } {RTABLE M7R0 I6RTABLE_SAVE/145473808X,%)anythingG6"6"[gl!"%!!!#%"#"#"")"#9""("#:6" } {RTABLE M7R0 I6RTABLE_SAVE/145477876X,%)anythingG6"6"[gl!"%!!!#%"#"#""""#9""!"#A6" } {RTABLE M7R0 I6RTABLE_SAVE/145480648X,%)anythingG6"6"[gl!"%!!!#%"#"#"""""$""#""%6" } {RTABLE M7R0 I6RTABLE_SAVE/145482052X,%)anythingG6"6"[gl!"%!!!#%"#"#!"#""&"""""$6" } {RTABLE M7R0 I6RTABLE_SAVE/145484752X,%)anythingG6"6"[gl!"%!!!#%"#"#""$""%""("""6" } {RTABLE M7R0 I6RTABLE_SAVE/145486444X,%)anythingG6"6"[gl!"%!!!#%"#"#"")"#9""("#:6" } {RTABLE M7R0 I6RTABLE_SAVE/145489216X,%)anythingG6"6"[gl!"%!!!#%"#"#"#_"$-""#j"$8"6" } {RTABLE M7R0 I6RTABLE_SAVE/145490188X,%)anythingG6"6"[gl!"%!!!#%"#"#!"#"#F!#8"#Q6" } {RTABLE M7R0 I6RTABLE_SAVE/145493032X,%)anythingG6"6"[gl!"%!!!#%"#"#"#_"$-""#j"$8"6" } {RTABLE M7R0 I6RTABLE_SAVE/145494580X,%)anythingG6"6"[gl!"%!!!#%"#"#"""""$""#""%6" } {RTABLE M7R0 I6RTABLE_SAVE/145498972X,%)anythingG6"6"[gl!"%!!!#%"#"#!"#""&"""""$6" } {RTABLE M7R0 I6RTABLE_SAVE/145500448X,%)anythingG6"6"[gl!"%!!!#%"#"#""$""%""("""6" } {RTABLE M7R0 I6RTABLE_SAVE/145505740X,%)anythingG6"6"[gl!"%!!!#%"#"#"""""*"")""%6" } {RTABLE M7R0 I6RTABLE_SAVE/145507216X,%)anythingG6"6"[gl!"%!!!#%"#"#"#>"#R"#;"#S6" } {RTABLE M7R0 I6RTABLE_SAVE/145509916X,%)anythingG6"6"[gl!"%!!!#%"#"#"")"#9""("#:6" } {RTABLE M7R0 I6RTABLE_SAVE/145511320X,%)anythingG6"6"[gl!"%!!!#%"#"#"#6"#D""*F(6" } {RTABLE M7R0 I6RTABLE_SAVE/145514020X,%)anythingG6"6"[gl!"%!!!#%"#"#"#>"#R"#;"#S6" } {RTABLE M7R0 I6RTABLE_SAVE/145515424X,%)anythingG6"6"[gl!"%!!!#%"#"#%"aG%"cG%"bG%"dG6" } {RTABLE M7R0 I6RTABLE_SAVE/145518196X,%)anythingG6"6"[gl!"%!!!#%"#"#%"pG%"rG%"qG%"sG6" } {RTABLE M7R0 I6RTABLE_SAVE/145521292X,%)anythingG6"6"[gl!"%!!!#%"#"#%"tG%"vG%"uG%"wG6" } {RTABLE M7R0 I6RTABLE_SAVE/145524280X,%)anythingG6"6"[gl!"%!!!#%"#"#,&%"pG"""%"tGF),&%"rGF)% "vGF),&%"qGF)%"uGF),&%"sGF)%"wGF)6" } {RTABLE M7R0 I6RTABLE_SAVE/145528348X,%)anythingG6"6"[gl!"%!!!#%"#"#,&*&%"aG""",&%"pGF*%"tGF *F*F**&%"bGF*,&%"rGF*%"vGF*F*F*,&*&%"cGF*F+F*F**&%"dGF*F0F*F*,&*&F)F*,&%"qGF*%" uGF*F*F**&F/F*,&%"sGF*%"wGF*F*F*,&*&F5F*F:F*F**&F7F*F>F*F*6" } {RTABLE M7R0 I6RTABLE_SAVE/145532992X,%)anythingG6"6"[gl!"%!!!#%"#"#,&*&%"aG"""%"pGF*F**&%"b GF*%"rGF*F*,&*&%"cGF*F+F*F**&%"dGF*F.F*F*,&*&F)F*%"qGF*F**&F-F*%"sGF*F*,&*&F1F* F6F*F**&F3F*F8F*F*6" } {RTABLE M7R0 I6RTABLE_SAVE/145534468X,%)anythingG6"6"[gl!"%!!!#%"#"#,&*&%"aG"""%"tGF*F**&%"b GF*%"vGF*F*,&*&%"cGF*F+F*F**&%"dGF*F.F*F*,&*&F)F*%"uGF*F**&F-F*%"wGF*F*,&*&F1F* F6F*F**&F3F*F8F*F*6" } {RTABLE M7R0 I6RTABLE_SAVE/145537312X,%)anythingG6"6"[gl!"%!!!#%"#"#,**&%"aG"""%"pGF*F**&%"b GF*%"rGF*F**&F)F*%"tGF*F**&F-F*%"vGF*F*,**&%"cGF*F+F*F**&%"dGF*F.F*F**&F5F*F0F* F**&F7F*F2F*F*,**&F)F*%"qGF*F**&F-F*%"sGF*F**&F)F*%"uGF*F**&F-F*%"wGF*F*,**&F5F *FF*F**&F5F*F@F*F**&F7F*FBF*F*6" }