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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 }} {SECT 0 {EXCHG {PARA 260 "" 0 "" {TEXT 286 39 "MATRICES AND MATRIX OPE RATIONS: Unit 18" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{PARA 261 "" 0 "" {TEXT 288 23 "Dr. Wlodzislaw Kostecki" }}{PARA 262 "" 0 " " {TEXT -1 53 "The Papua New Guinea University of Technology (PNGUT)" }}{PARA 263 "" 0 "" {TEXT -1 54 "Department of Electrical and Communic ation Engineering" }}{PARA 264 "" 0 "" {TEXT -1 20 "Lae, Morobe Provin ce" }}{PARA 265 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 266 "" 0 "" {TEXT 287 41 "Copyright \251 200 0 by Wlodzislaw Kostecki" }}{PARA 267 "" 0 "" {TEXT 289 19 "All right s reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 "" {TEXT 290 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 257 4 "(18)" }{TEXT 268 1 " " }{TEXT 267 51 "Replacing a colum n in a matrix with a column matrix" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 308 10 "OBJECTIVES" }{TEXT 309 1 ":" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 311 1 "\225" }{TEXT -1 112 " To introduce this operation as a basis for solving systems of linear algebraic inhomogeneous equa tions using " }{TEXT 312 7 "Cramer\222" }{TEXT -1 1 "s" }{TEXT 313 2 " " }{TEXT -1 5 "rule." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 310 1 "\225" }{TEXT -1 88 " To provide alternative methods of replacing a column i n a matrix with a column matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 314 1 "\225" }{TEXT -1 131 " To show how to carry over the original matri x unchanged for use in computations that may follow its column-replace ment operation." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "restart : with(linalg, copyinto, \+ rowdim) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 276 1 "(" } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 258 3 " \327 " }{XPPEDIT 18 0 "3" " 6#\"\"$" }{TEXT 277 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 259 1 "A" } {TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "A := matrix(3, 3, [a[11], a[12], a[13], a[21], a[22], a[23], a[31], a [32], a[33]]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %\"AG-%'matrixG6#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 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "and a " }{TEXT 278 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 279 3 " \327 " } {XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 280 1 ")" }{TEXT -1 17 " column m atrix [" }{TEXT 260 2 "CM" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "CM := matrix(3, 1, [cm[11], cm[21], cm[31 ]]) : CM = matrix(CM) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#CMG-%' matrixG6#7%7#&%#cmG6#\"#67#&F+6#\"#@7#&F+6#\"#J" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Replace the \+ " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT -1 12 "nd column " }{TEXT 281 2 "( " }{XPPEDIT 18 0 "j = 2" "6#/%\"jG\"\"#" }{TEXT 282 2 " )" } {TEXT -1 17 " of the matrix [" }{TEXT 261 1 "A" }{TEXT -1 42 "] with \+ the elements of the column matrix [" }{TEXT 262 2 "CM" }{TEXT -1 8 "]. This " }{TEXT 296 8 "modifies" }{TEXT -1 10 " matrix [" }{TEXT 264 1 "A" }{TEXT -1 21 "] and returns a new " }{TEXT 283 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 284 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 285 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 263 1 "B" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 304 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 294 4 "N.B." }{TEXT -1 31 " Since replac ing a column in [" }{TEXT 295 1 "A" }{TEXT -1 38 "] changes it and the original matrix [" }{TEXT 297 1 "A" }{TEXT -1 96 "] may be needed in \+ further computations, it is convenient to create its copy at this poin t, e.g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "C_A := copy(A) \+ : C_A = matrix(C_A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$C_AG-%'ma trixG6#7%7%&%\"aG6#\"#6&F+6#\"#7&F+6#\"#87%&F+6#\"#@&F+6#\"#A&F+6#\"#B 7%&F+6#\"#J&F+6#\"#K&F+6#\"#L" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "Maintaining the original mat rix is necessary, for example, in solving systems of linear algebraic \+ inhomogeneous equations by means of " }{TEXT 303 7 "Cramer\222" } {TEXT -1 29 "s rule \226 refer to Unit (19)." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 271 "" 0 "" {TEXT 305 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "The column-replacement operation may be performed using any of \+ the following alternative methods." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 8 "Method 1" }{TEXT -1 12 ". Using the " }{TEXT 275 3 "for" }{TEXT -1 16 "-loop construct:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "j := 2 : k := 1 : for i to rowdim(A) do A[i,j] := CM[i,k] od : B := matrix(A) : B = mat rix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7%7%&%\" aG6#\"#6&%#cmGF,&F+6#\"#87%&F+6#\"#@&F/F5&F+6#\"#B7%&F+6#\"#J&F/F=&F+6 #\"#L" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Restore the original matrix [" }{TEXT 298 1 "A" } {TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "A := matr ix(C_A) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG- %'matrixG6#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 0 "" 0 "" {TEXT 270 8 "Method 2" }{TEXT -1 12 ". \+ Using the " }{TEXT 271 8 "copyinto" }{TEXT -1 10 " function:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "B := copyinto(CM, A, k, j) \+ : B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG 6#7%7%&%\"aG6#\"#6&%#cmGF,&F+6#\"#87%&F+6#\"#@&F/F5&F+6#\"#B7%&F+6#\"# J&F/F=&F+6#\"#L" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Restore the original matrix [" }{TEXT 299 1 "A" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "A := matrix(C_A) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#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 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 8 "Method 3" } {TEXT -1 12 ". Using the " }{TEXT 273 4 "subs" }{TEXT -1 10 " function :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "B := subs(a[12]=cm[11] , a[22]=cm[21], a[32]=cm[31], matrix(A)) : B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7%7%&%\"aG6#\"#6&%#cmGF,&F +6#\"#87%&F+6#\"#@&F/F5&F+6#\"#B7%&F+6#\"#J&F/F=&F+6#\"#L" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 301 4 "N .B." }{TEXT -1 19 " This method does " }{TEXT 300 3 "not" }{TEXT -1 29 " affect the original matrix [" }{TEXT 302 1 "A" }{TEXT -1 7 "], i. e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "A = matrix(A) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#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 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 " " {TEXT 265 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 266 4 "N.B." }{TEXT -1 160 " The operati on of replacing a column in a matrix with a column matrix is necessary in solving systems of linear algebraic inhomogeneous equations by mea ns of " }{TEXT 269 7 "Cramer\222" }{TEXT -1 29 "s rule \226 refer to Unit (19)." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 269 "" 0 "" {TEXT 293 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Proceed to Unit (19) for \"" }{TEXT 292 76 "Solution of systems of linear algebraic inhomogeneous e quations using Cramer" }{TEXT 306 1 "\222" }{TEXT 307 6 "s rule" } {TEXT -1 2 "\"." }}}{EXCHG {PARA 259 "" 0 "" {TEXT 291 67 "----------- --------------------------------------------------------" }}}}{MARK "4 0" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }