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}3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 264 "" 0 "" {TEXT 347 38 "MATRICES AND MATRIX OPE RATIONS: Unit 8" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{PARA 265 "" 0 "" {TEXT 349 23 "Dr. Wlodzislaw Kostecki" }}{PARA 266 "" 0 "" {TEXT -1 53 "The Papua New Guinea University of Technology (PNGUT)" }} {PARA 267 "" 0 "" {TEXT -1 54 "Department of Electrical and Communicat ion Engineering" }}{PARA 268 "" 0 "" {TEXT -1 20 "Lae, Morobe Province " }}{PARA 269 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 " " {TEXT -1 0 "" }}{PARA 270 "" 0 "" {TEXT 348 41 "Copyright \251 2000 by Wlodzislaw Kostecki" }}{PARA 271 "" 0 "" {TEXT 350 19 "All rights reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 272 "" 0 "" {TEXT 351 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 257 3 "(8)" }{TEXT 298 1 " " }{TEXT 297 38 "Multiplication of \+ a matrix by a number" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 355 10 "OBJECTIVES" }{TEXT 356 1 ":" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 358 1 "\225" }{TEXT -1 68 " To define the operation of multiplication of a matrix by a number." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 357 1 "\225 " }{TEXT -1 80 " To provide alternative methods of multiplication of \+ a matrix by a number with " }{TEXT 359 5 "Maple" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "restart : with(linalg, diag, multiply, scalarmul) : " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Multiplication of a matrix by a number is sometimes calle d multiplication of a matrix by a " }{TEXT 360 6 "scalar" }{TEXT -1 102 " to distinguish it from multiplication by another matrix. The sc alar may be a real or complex number." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "If an " }{TEXT 320 1 "(" }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 321 3 " \327 " } {XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT 322 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 258 1 "A" }{TEXT -1 29 "] is multiplied by a scalar " } {XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 12 ", whether " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 72 " is real or complex, then each element \+ in the matrix is multiplied by " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 323 1 "(" } {XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 324 3 " \327 " }{XPPEDIT 18 0 "2" " 6#\"\"#" }{TEXT 325 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 270 1 "A" } {TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 " A := matrix(4, 2, [a, b, c, d, e, f, g, h]) : A = matrix(A) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7&7$%\"aG%\"bG7$%\"c G%\"dG7$%\"eG%\"fG7$%\"gG%\"hG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The product " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 2 " [" }{TEXT 282 1 "A" }{TEXT -1 66 "] can be computed using any of the following alternative methods." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 299 8 "Method 1" }{TEXT -1 12 ". Using the " }{TEXT 300 9 "scalarmul" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "`kA` := scalarmul(A, k) : k*A = matrix(`kA`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"kG\"\"\"%\"AGF&-%'matrixG6#7&7$*&F%F&%\"aGF&*&F%F &%\"bGF&7$*&F%F&%\"cGF&*&F%F&%\"dGF&7$*&F%F&%\"eGF&*&F%F&%\"fGF&7$*&F% F&%\"gGF&*&F%F&%\"hGF&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 301 8 "Method 2" }{TEXT -1 12 ". Using th e " }{TEXT 302 5 "evalm" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 45 "`kA` := evalm(k * A) : k*A = matrix(`kA`) \+ ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"kG\"\"\"%\"AGF&-%'matrixG6# 7&7$*&F%F&%\"aGF&*&F%F&%\"bGF&7$*&F%F&%\"cGF&*&F%F&%\"dGF&7$*&F%F&%\"e GF&*&F%F&%\"fGF&7$*&F%F&%\"gGF&*&F%F&%\"hGF&" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 262 "" 0 "" {TEXT 315 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 308 4 "N.B." }{TEXT -1 20 " The order of the " }{TEXT 306 10 "multip lier" }{TEXT -1 7 " and " }{TEXT 307 12 "multiplicand" }{TEXT -1 9 " in the " }{TEXT 309 5 "evalm" }{TEXT -1 15 " function does " }{TEXT 310 3 "not" }{TEXT -1 48 " matter for this type of multiplication. If \+ the " }{TEXT 312 9 "scalarmul" }{TEXT -1 22 " function is used for " } {TEXT 313 6 "scalar" }{TEXT -1 1 " " }{TEXT 314 3 "mul" }{TEXT -1 41 " tiplication, the order of the arguments " }{TEXT 19 4 "must" }{TEXT -1 11 " be this: " }{TEXT 311 16 "(matrix, scalar)" }{TEXT -1 1 "." } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 263 "" 0 "" {TEXT 316 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 303 8 "Method 3" }{TEXT -1 12 ". Using the " } {TEXT 304 3 "map" }{TEXT -1 60 " function together with the arrow-type functional operator " }{TEXT 326 2 "( " }{TEXT 305 3 "x->" }{TEXT 327 2 " )" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "`kA` := map(x->x*k, A) : k*A = matrix(`kA`) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*&%\"kG\"\"\"%\"AGF&-%'matrixG6#7&7$*&F%F&%\"aGF&*&F %F&%\"bGF&7$*&F%F&%\"cGF&*&F%F&%\"dGF&7$*&F%F&%\"eGF&*&F%F&%\"fGF&7$*& F%F&%\"gGF&*&F%F&%\"hGF&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "This scalar multiplication of a ma trix may be displayed in \"like-in-a-book\" form, namely" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "k * matrix(A) = matrix(`kA`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"kG\"\"\"-%'matrixG6#7&7$%\"aG%\" bG7$%\"cG%\"dG7$%\"eG%\"fG7$%\"gG%\"hGF&-F(6#7&7$*&F%F&F,F&*&F%F&F-F&7 $*&F%F&F/F&*&F%F&F0F&7$*&F%F&F2F&*&F%F&F3F&7$*&F%F&F5F&*&F%F&F6F&" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 287 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 290 4 "N.B." }{TEXT -1 46 " An essential differ ence arises in using the " }{TEXT 288 5 "evalm" }{TEXT -1 5 " and " } {TEXT 289 9 "scalarmul" }{TEXT -1 17 " functions when " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 74 " is the multiplier of a matrix, the former \+ function yielding the scalar " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 17 ", the latter a " }{TEXT 361 4 "zero" }{TEXT -1 9 " matrix." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "For example, consider the " }{TEXT 328 1 "(" }{XPPEDIT 18 0 "4 " "6#\"\"%" }{TEXT 329 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 330 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 291 1 "A" }{TEXT -1 28 "] u sed earlier in this Unit." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "`0A` := evalm(0 &* A) : `0 A` = `0A` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$0~AG\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`0A` := scalarmul(A, 0) : `0 A` = matrix(`0A`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$0~AG-%' matrixG6#7&7$\"\"!F*F)F)F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 260 "" 0 "" {TEXT 284 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "It seems to be instructive to see how multiplication of a matrix by a scalar could b e executed with " }{TEXT 271 5 "Maple" }{TEXT -1 8 " if the " }{TEXT 272 9 "scalarmul" }{TEXT 317 1 "," }{TEXT -1 1 " " }{TEXT 283 5 "evalm " }{TEXT 318 1 "," }{TEXT -1 5 " and " }{TEXT 319 3 "map" }{TEXT -1 16 " functions were " }{TEXT 285 3 "not" }{TEXT -1 11 " available." }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "To perform this multiplication with " }{TEXT 273 5 "Maple" } {TEXT -1 14 ", the scalar " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 38 " must be presented in the form of a " }{TEXT 259 13 "scalar matr ix" }{TEXT -1 3 " [" }{TEXT 267 1 "K" }{TEXT -1 12 "] of order " } {TEXT 331 1 "(" }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 332 3 " \327 " } {XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 333 1 ")" }{TEXT -1 95 ". Accordin g to the multiplication conformability rule, the number of columns and rows in the " }{TEXT 261 13 "scalar matrix" }{TEXT -1 15 " (which i s a " }{TEXT 260 8 "diagonal" }{TEXT -1 10 ", hence " }{TEXT 262 6 "square" }{TEXT -1 54 " matrix) must be equal to the number of rows i n the " }{TEXT 268 15 "post-multiplied" }{TEXT -1 2 " " }{TEXT 334 1 "(" }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 335 3 " \327 " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT 336 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 263 1 "A" }{TEXT -1 7 "], or " }{XPPEDIT 18 0 "m=n" "6#/%\"mG%\"nG" } {TEXT -1 32 ". The product matrix is a new " }{TEXT 337 1 "(" } {XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 338 3 " \327 " }{XPPEDIT 18 0 "p" " 6#%\"pG" }{TEXT 339 1 ")" }{TEXT -1 37 " matrix, every element of whi ch is " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 37 " times that of th e original matrix [" }{TEXT 265 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "T herefore, the " }{TEXT 266 15 "pre-multiplying" }{TEXT -1 17 " scala r matrix [" }{TEXT 269 1 "K" }{TEXT -1 20 "] must be of order " } {TEXT 340 1 "(" }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 341 3 " \327 " } {XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 342 1 ")" }{TEXT -1 7 ", viz." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "K := diag(k, k, k, k) : K \+ = matrix(K) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"KG-%'matrixG6#7&7 &%\"kG\"\"!F+F+7&F+F*F+F+7&F+F+F*F+7&F+F+F+F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Thus, the prod uct " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 2 " [" }{TEXT 264 1 "A" }{TEXT -1 54 "] is obtained as a result of matrix multiplication [" }{TEXT 274 1 "K" }{TEXT -1 3 "] [" }{TEXT 275 1 "A" }{TEXT -1 8 "], v iz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "`kA` := multiply(K, \+ A) : k*A = matrix(`kA`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"k G\"\"\"%\"AGF&-%'matrixG6#7&7$*&F%F&%\"aGF&*&F%F&%\"bGF&7$*&F%F&%\"cGF &*&F%F&%\"dGF&7$*&F%F&%\"eGF&*&F%F&%\"fGF&7$*&F%F&%\"gGF&*&F%F&%\"hGF& " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "or," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "matrix(K ) * matrix(A) = matrix(`kA`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&- %'matrixG6#7&7&%\"kG\"\"!F+F+7&F+F*F+F+7&F+F+F*F+7&F+F+F+F*\"\"\"-F&6# 7&7$%\"aG%\"bG7$%\"cG%\"dG7$%\"eG%\"fG7$%\"gG%\"hGF/-F&6#7&7$*&F*F/F4F /*&F*F/F5F/7$*&F*F/F7F/*&F*F/F8F/7$*&F*F/F:F/*&F*F/F;F/7$*&F*F/F=F/*&F *F/F>F/" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This operation may be displayed in \"like-in-a-book \" form, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "k * matrix (A) = matrix(`kA`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"kG\"\"\" -%'matrixG6#7&7$%\"aG%\"bG7$%\"cG%\"dG7$%\"eG%\"fG7$%\"gG%\"hGF&-F(6#7 &7$*&F%F&F,F&*&F%F&F-F&7$*&F%F&F/F&*&F%F&F0F&7$*&F%F&F2F&*&F%F&F3F&7$* &F%F&F5F&*&F%F&F6F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 261 "" 0 "" {TEXT 292 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 286 60 "Numerical exam ple for multiplication of a matrix by a number" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Let the elem ents of the matrix [" }{TEXT 276 1 "A" }{TEXT -1 61 "] defined above b e now assigned numerical values, as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "a:=-1 : b:=2 : c:=1 : d:=3 : e:=3 : f:=-1 : g:=- 2 : h:=2 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "A := map(x->e val(x), A) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\" AG-%'matrixG6#7&7$!\"\"\"\"#7$\"\"\"\"\"$7$F.F*7$!\"#F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "L et the multiplier number be " }{XPPEDIT 18 0 " k=2" "6#/%\"kG\"\"#" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "k := 2 :" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "(a) Multiplication using the " }{TEXT 293 9 "scalarmul" } {TEXT -1 5 " and " }{TEXT 294 5 "evalm" }{TEXT -1 11 " functions:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`kA` := scalarmul(A, k) : \+ k * A = matrix(`kA`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$%\"AG\"\" #-%'matrixG6#7&7$!\"#\"\"%7$F&\"\"'7$F/F,7$!\"%F-" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "`kA` := evalm(k * A) : k * A = matrix(`kA`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$%\"AG\"\"#-%'matrixG6#7&7$!\"#\"\"%7$F&\"\"'7$F/F,7$ !\"%F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "(b) Multiplication without using the " }{TEXT 295 9 "scalarmul" }{TEXT -1 5 " and " }{TEXT 296 5 "evalm" }{TEXT -1 11 " fu nctions:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 58 "According to the multiplication conformability rul e, the " }{TEXT 277 15 "pre-multiplying" }{TEXT -1 17 " scalar matri x [" }{TEXT 278 1 "K" }{TEXT -1 20 "] must be of order " }{TEXT 343 1 "(" }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 344 3 " \327 " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 345 1 ")" }{TEXT -1 19 " and have the form" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "k := 'k' : K := subs(k=2, matrix(K)) : K = matrix(K) ; k := 2 :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"KG-%'matrixG6#7&7&\"\"#\"\"!F+F+7&F+F*F+F+7&F+F+F*F +7&F+F+F+F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Thus, the product " }{XPPEDIT 18 0 "2" "6#\"\" #" }{TEXT -1 1 "[" }{TEXT 279 1 "A" }{TEXT -1 1 "]" }{TEXT 346 3 " = \+ " }{TEXT -1 1 "[" }{TEXT 280 1 "K" }{TEXT -1 3 "] [" }{TEXT 281 1 "A" }{TEXT -1 5 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`kA` \+ := multiply(K, A) : k * A = matrix(`kA`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$%\"AG\"\"#-%'matrixG6#7&7$!\"#\"\"%7$F&\"\"'7$F/F,7$ !\"%F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "or, in \"like-in-a-book\" form," }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 30 "k * matrix(A) = matrix(`kA`) ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,$-%'matrixG6#7&7$!\"\"\"\"#7$\"\"\"\"\"$7$F.F* 7$!\"#F+F+-F&6#7&7$F1\"\"%7$F+\"\"'7$F8F17$!\"%F6" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 273 "" 0 "" {TEXT 354 5 "* * *" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Proceed to Unit (9) for \"" }{TEXT 353 40 "Multiplication involving the unit matrix" }{TEXT -1 2 "\"." }}}{EXCHG {PARA 258 "" 0 "" {TEXT 352 67 "--------------------------------------------------- ----------------" }}}}{MARK "53 0 0" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }