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Output" -1 12 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Arial Narrow " 1 12 128 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 128 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Norma l" -1 258 1 {CSTYLE "" -1 -1 "Arial Narrow" 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" -1 259 1 {CSTYLE "" -1 -1 "Helvetica" 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 "Normal" -1 260 1 {CSTYLE "" -1 -1 "Ari al Narrow" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 324 39 "MATRICES AND MATRIX OPE RATIONS: Unit 12" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{PARA 258 "" 0 "" {TEXT 326 23 "Dr. Wlodzislaw Kostecki" }}{PARA 260 "" 0 " " {TEXT -1 53 "The Papua New Guinea University of Technology (PNGUT)" }}{PARA 260 "" 0 "" {TEXT -1 54 "Department of Electrical and Communic ation Engineering" }}{PARA 260 "" 0 "" {TEXT -1 20 "Lae, Morobe Provin ce" }}{PARA 260 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 325 41 "Copyright \251 200 0 by Wlodzislaw Kostecki" }}{PARA 258 "" 0 "" {TEXT 327 19 "All right s reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 328 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 257 4 "(12)" }{TEXT 310 1 " " }{TEXT 309 42 "The minor and cof actor of a matrix element" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 340 10 "OBJECTIVES" }{TEXT 341 1 ":" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 342 1 "\225" }{TEXT -1 17 " To define the " }{TEXT 343 5 "minor" } {TEXT -1 7 " and " }{TEXT 344 8 "cofactor" }{TEXT -1 22 " of a matr ix element." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 345 1 "\225" }{TEXT -1 69 " To provide alternative methods of computing the minor and cofact or." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 346 1 "\225" }{TEXT -1 20 " To i ntroduce the " }{TEXT 347 7 "Laplace" }{TEXT -1 45 " expansion theor em and show its application." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 348 1 " \225" }{TEXT -1 38 " To introduce further functions from " }{TEXT 350 5 "Maple" }{TEXT 349 1 "\222" }{TEXT -1 3 "s " }{TEXT 351 4 "main " }{TEXT -1 49 " library that are useful in matrix computations." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "restart : with(linalg, coldim, delcols, delrows, de t, minor, rowdim) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Both the minor and cofactor of a m atrix element are defined " }{TEXT 365 4 "only" }{TEXT -1 21 " for squ are matrices." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Consider a " }{TEXT 313 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 264 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 314 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 265 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "A := mat rix(3, 3, [a[11], a[12], a[13], a[21], a[22], a[23], a[31], a[32], a[3 3]]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'m atrixG6#7%7%&%\"aG6#\"#6&F+6#\"#7&F+6#\"#87%&F+6#\"#@&F+6#\"#A&F+6#\"# B7%&F+6#\"#J&F+6#\"#K&F+6#\"#L" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 3 "A. " }{TEXT 307 29 "The mino r of a matrix element" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 262 5 "minor" }{TEXT -1 17 " of an element " }{XPPEDIT 18 0 "a[ij]" "6#&%\"aG6#%#ijG" } {TEXT -1 9 " in an " }{TEXT 315 1 "(" }{XPPEDIT 18 0 "n" "6#%\"nG" } {TEXT 316 3 " \327 " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 317 1 ")" } {TEXT -1 28 " matrix is defined as the " }{TEXT 300 6 "matrix" } {TEXT -1 12 " of order " }{TEXT 318 1 "(" }{XPPEDIT 18 0 "n-1" "6#,& %\"nG\"\"\"F%!\"\"" }{TEXT 319 5 ") \327 (" }{XPPEDIT 18 0 "n-1" "6#,& %\"nG\"\"\"F%!\"\"" }{TEXT 320 1 ")" }{TEXT -1 35 " obtained by the d eletion of row " }{XPPEDIT 18 0 "i" "6#%\"iG" }{TEXT -1 14 " and col umn " }{XPPEDIT 18 0 "j" "6#%\"jG" }{TEXT -1 1 "." }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "For example, choose the e lement " }{XPPMATH 20 "6#&%\"aG6#\"#7" }{TEXT -1 14 " in the row " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 14 " and column " } {XPPEDIT 18 0 "j=2" "6#/%\"jG\"\"#" }{TEXT -1 17 " of the matrix [" } {TEXT 266 1 "A" }{TEXT -1 18 "] and obtain the " }{TEXT 263 5 "minor " }{TEXT -1 101 " of this element. This operation may be performed us ing either of the following alternative methods." }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 311 8 "Method 1" }{TEXT -1 12 ". Using the " }{TEXT 304 7 "delrows" }{TEXT -1 5 " and \+ " }{TEXT 305 7 "delcols" }{TEXT -1 11 " functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "`minor(a12)` := delcols(delrows(A, 1..1), 2 ..2) : Minor(a[12]) = matrix(`minor(a12)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&MinorG6#&%\"aG6#\"#7-%'matrixG6#7$7$&F(6#\"#@&F(6# \"#B7$&F(6#\"#J&F(6#\"#L" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 312 8 "Method 2" }{TEXT -1 12 ". Using th e " }{TEXT 306 5 "minor" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 93 "i := 1 : j := 2 : `minor(a12)` := minor(A , i, j) : Minor(a[12]) = matrix(`minor(a12)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&MinorG6#&%\"aG6#\"#7-%'matrixG6#7$7$&F(6#\"#@&F(6# \"#B7$&F(6#\"#J&F(6#\"#L" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 287 11 "determinant" }{TEXT -1 10 " of the " }{TEXT 288 5 "minor" }{TEXT -1 18 " of the \+ element " }{XPPMATH 20 "6#&%\"aG6#\"#7" }{TEXT -1 4 " is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "`det(minor(a12))` := det(`minor(a12 )`) : Det(Minor(a[12])) = `det(minor(a12))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#-%&MinorG6#&%\"aG6#\"#7,&*&&F+6#\"#@\"\"\"&F+ 6#\"#LF3F3*&&F+6#\"#BF3&F+6#\"#JF3!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 3 "B. " }{TEXT 308 32 "The cofactor of a matrix element" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 258 8 "cof actor" }{TEXT -1 6 " or " }{TEXT 259 12 "signed minor" }{TEXT -1 17 " of an element " }{XPPEDIT 18 0 "a[ij]" "6#&%\"aG6#%#ijG" }{TEXT -1 9 " in an " }{TEXT 321 1 "(" }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 322 3 " \327 " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 323 1 ")" }{TEXT -1 43 " matrix is defined as the product of the " }{TEXT 301 11 "det erminant" }{TEXT -1 10 " of the " }{TEXT 284 5 "minor" }{TEXT -1 23 " of this element and " }{TEXT 285 6 "scalar" }{TEXT -1 2 " " } {XPPEDIT 18 0 "(-1)^(i+j)" "6#),$\"\"\"!\"\",&%\"iGF%%\"jGF%" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "For example, the " }{TEXT 286 8 "cofactor" }{TEXT -1 24 " of the above element " }{XPPMATH 20 "6#&%\"aG6#\"#7" }{TEXT -1 4 " is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "cofactor(a12) := (-1)^(i+j) \+ * `det(minor(a12))` : Cofactor(a[12]) = cofactor(a12) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)CofactorG6#&%\"aG6#\"#7,&*&&F(6#\"#@\"\"\"& F(6#\"#LF0!\"\"*&&F(6#\"#BF0&F(6#\"#JF0F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Using the " } {TEXT 267 9 "cofactors" }{TEXT -1 111 " of row or column elements of \+ a matrix enables evaluation of the determinant of the matrix, accordin g to the " }{TEXT 271 7 "Laplace" }{TEXT -1 34 " expansion theorem, \+ which states:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "\"The determinant associated with any " }{TEXT 353 1 "(" }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 354 3 " \327 " } {XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 355 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 352 1 "A" }{TEXT -1 108 "] is obtained by summing the product s of the elements and their cofactors in any row or column of the matr ix" }{TEXT 270 1 "." }{TEXT -1 1 "\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "If matrix [" }{TEXT 356 1 "A" }{TEXT -1 27 "] has the general element " }{XPPEDIT 18 0 "a [ij]" "6#&%\"aG6#%#ijG" }{TEXT -1 37 " and the corresponding cofactor is " }{XPPEDIT 18 0 "Cofactor(a[ij])" "6#-%)CofactorG6#&%\"aG6#%#ijG " }{TEXT -1 49 ", then this theorem may be expressed as follows:" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 357 1 "\225" }{TEXT -1 33 " Expansion by elements of a row:" }}} {EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "Det(A) = Sum(a[ij]*Cofactor(a[ ij]), j=1..n)" "6#/-%$DetG6#%\"AG-%$SumG6$*&&%\"aG6#%#ijG\"\"\"-%)Cofa ctorG6#&F-6#F/F0/%\"jG;F0%\"nG" }{TEXT -1 10 " " }{TEXT 358 1 "i" }{TEXT 359 14 " = 1, 2, ..., " }{TEXT 360 1 "n" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 361 1 "\225" } {TEXT -1 36 " Expansion by elements of a column:" }}}{EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "Det(A) = Sum(a[ij]*Cofactor(a[ij]), i=1..n) " "6#/-%$DetG6#%\"AG-%$SumG6$*&&%\"aG6#%#ijG\"\"\"-%)CofactorG6#&F-6#F /F0/%\"iG;F0%\"nG" }{TEXT -1 10 " " }{TEXT 362 1 "j" }{TEXT 363 14 " = 1, 2, ..., " }{TEXT 364 1 "n" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 302 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 4 "N.B." }{TEXT -1 13 " Since the " }{TEXT 283 9 "expansion" } {TEXT -1 141 " of the determinant of a matrix can be done about any o ne row or any one column, it is convenient to choose a row or a column with as many " }{XPPMATH 20 "6#%&zerosG" }{TEXT -1 14 " as possible ." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 " " {TEXT 303 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Choosing the first row ( " } {XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 25 " ) of the above ma trix [" }{TEXT 272 1 "A" }{TEXT -1 60 "] results in the following expa nsion of the determinant of [" }{TEXT 273 1 "A" }{TEXT -1 37 "] displa yed in \"like-in-a-book\" form:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "i:=1 : for j to coldim(A) do `det(minor(A,i,j))`[i, j] := D et(minor(A, i, j)) : a[i, j] := a[cat(i, j)] : od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "j := 'j' : `det(A)` := sum((-1)^ (i+j)*a[i, j]*`det(minor(A,i,j))`[i, j], j=1..coldim(A)) : Det(A)=`d et(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG,(*&&%\"aG6 #%#11G\"\"\"-F%6#-%'matrixG6#7$7$&F+6#\"#A&F+6#\"#B7$&F+6#\"#K&F+6#\"# LF.F.*&&F+6#%#12GF.-F%6#-F26#7$7$&F+6#\"#@F97$&F+6#\"#JF@F.!\"\"*&&F+6 #%#13GF.-F%6#-F26#7$7$FMF67$FQF=F.F." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 294 5 "The " }{TEXT 289 3 "ca t" }{TEXT 295 29 " function is used above to " }{TEXT 290 1 "c" } {TEXT -1 3 "onc" }{TEXT 291 2 "at" }{TEXT -1 5 "enate" }{TEXT 296 40 " together the indices of the elements " }{XPPEDIT 18 0 "a[ij]" "6#&% \"aG6#%#ijG" }{TEXT 298 33 " into a string expression. The " }{TEXT 293 3 "sum" }{TEXT 299 25 " function performs the " }{TEXT 292 3 "su m" }{TEXT -1 6 "mation" }{TEXT 297 40 " of the expressions under this function" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Evaluation of the determinant of m atrix [" }{TEXT 282 1 "A" }{TEXT -1 34 "] yields the following express ion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "for j to coldim(A) \+ do `det(minor(A,i,j))`[i, j] := det(minor(A, i, j)) : od :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "j := 'j' : `det(A)` := su m((-1)^(i+j)*a[i, j]*`det(minor(A,i,j))`[i, j], j=1..coldim(A)) : De t(A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG,( *&&%\"aG6#%#11G\"\"\",&*&&F+6#\"#AF.&F+6#\"#LF.F.*&&F+6#\"#BF.&F+6#\"# KF.!\"\"F.F.*&&F+6#%#12GF.,&*&&F+6#\"#@F.F4F.F.*&F8F.&F+6#\"#JF.F>F.F> *&&F+6#%#13GF.,&*&FEF.F;F.F.*&F1F.FIF.F>F.F." }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Application of \+ the " }{TEXT 276 6 "normal" }{TEXT -1 2 ", " }{TEXT 277 8 "simplify" } {TEXT -1 5 ", or " }{TEXT 278 6 "expand" }{TEXT -1 69 " function to th e above result expands it to the following expression:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "`det(A)` := normal(`det(A)`) : De t(A) = `det(A)` ; a[1,2] := 'a[1,2]' :" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG,.*(&%\"aG6#%#11G\"\"\"&F+6#\"#AF.&F+6#\"#LF.F.*( F*F.&F+6#\"#BF.&F+6#\"#KF.!\"\"*(&F+6#%#12GF.&F+6#\"#@F.F2F.F<*(F>F.F6 F.&F+6#\"#JF.F.*(&F+6#%#13GF.FAF.F9F.F.*(FIF.F/F.FEF.F<" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 280 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 4 "N.B." }{TEXT -1 47 " Evaluation of the determinant \+ of the matrix [" }{TEXT 274 1 "A" }{TEXT -1 17 "] can be done in " } {TEXT 279 5 "Maple" }{TEXT -1 1 " " }{TEXT 269 8 "directly" }{TEXT -1 17 " by applying the " }{TEXT 261 3 "det" }{TEXT -1 25 " function to t he matrix [" }{TEXT 275 1 "A" }{TEXT -1 7 "], i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "`det(A)` := sort(det(A)) : Det(A) = `det( A)` ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG,.*(&%\"aG6#\" #6\"\"\"&F+6#\"#KF.&F+6#\"#BF.!\"\"*(F*F.&F+6#\"#AF.&F+6#\"#LF.F.*(&F+ 6#\"#8F.F/F.&F+6#\"#@F.F.*(F>F.F7F.&F+6#\"#JF.F5*(&F+6#\"#7F.FAF.F:F.F 5*(FIF.F2F.FEF.F." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "The sorting function " }{TEXT 339 4 "sort " }{TEXT -1 86 " is used above to obtain a strict correspondence of th e product terms in both results." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 281 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 333 4 "N.B." } {TEXT -1 23 " It follows from the " }{TEXT 332 7 "Laplace" }{TEXT -1 162 " expansion theorem that the sum of the products of the elemen ts of any row (or column) of a square matrix with the cofactors corres ponding to the elements of a " }{TEXT 338 9 "different" }{TEXT -1 22 " row (or column) is " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 1 "." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 335 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 336 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 337 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 334 1 "A" }{TEXT -1 10 "] g iven as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "A := matrix(3, 3 , [1, 3, 2, 4, 1, 2, 3, 1, 3]) : A = matrix(A) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"\"\"\"$\"\"#7%\"\"%F*F,7%F+ F*F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 137 "and verify that the sum of the products of the element s of column 1 with the corresponding cofactors of the elements of co lumn 2 is " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "( a) The cofactors of the elements of column " }{XPPEDIT 18 0 "2" "6#\" \"#" }{TEXT -1 5 " are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "j := 2 : for i to rowdim(A) do cofactor(a[i, j]) := (-1)^(i+j)*det( minor(A, i, j)) : print(Cofactor(a[i, j]) = cofactor(a[i, j])) : o d : i := 'i' :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)CofactorG6#&% \"aG6$\"\"\"\"\"#!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)CofactorG 6#&%\"aG6$\"\"#F*!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)CofactorG 6#&%\"aG6$\"\"$\"\"#\"\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "(b) The sum of the products of the elements of column " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT -1 62 " w ith the corresponding cofactors of the elements of column " } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT -1 4 " is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`sum(A[i,1] cofactor(a[i,j])` := sum(A[i, 1] * c ofactor(a[i, j]), i=1..rowdim(A)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "Sum(a[i, 1] * Cofactor(a[i, j]), i=1..rowdim(A)) = `s um(A[i,1] cofactor(a[i,j])` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$S umG6$*&&%\"aG6$%\"iG\"\"\"F,-%)CofactorG6#&F)6$F+\"\"#F,/F+;F,\"\"$\" \"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 331 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Proceed to Unit (13) for \"" } {TEXT 330 23 "The adjoint of a matrix" }{TEXT -1 2 "\"." }}}{EXCHG {PARA 258 "" 0 "" {TEXT 329 67 "-------------------------------------- -----------------------------" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }