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1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 269 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 }} {SECT 0 {EXCHG {PARA 260 "" 0 "" {TEXT 306 38 "MATRICES AND MATRIX OPE RATIONS: Unit 6" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{PARA 261 "" 0 "" {TEXT 308 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 Communicat ion Engineering" }}{PARA 264 "" 0 "" {TEXT -1 20 "Lae, Morobe Province " }}{PARA 265 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 " " {TEXT -1 0 "" }}{PARA 266 "" 0 "" {TEXT 307 41 "Copyright \251 2000 by Wlodzislaw Kostecki" }}{PARA 267 "" 0 "" {TEXT 309 19 "All rights reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 "" {TEXT 310 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 257 3 "(6)" }{TEXT 274 1 " " }{TEXT 273 69 "Multiplication of \+ a multi-row multi-column matrix and a column matrix" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 314 10 "OBJECT IVES" }{TEXT 315 1 ":" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 317 1 "\225" }{TEXT -1 112 " To introduc e this operation as a basis for solving systems of linear algebraic in homogeneous equations using " }{TEXT 318 7 "Cramer\222" }{TEXT -1 1 " s" }{TEXT 319 2 " " }{TEXT -1 5 "rule." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 316 1 "\225" }{TEXT -1 95 " To provide examples of multiplicati on of a multi-row multi-column matrix and a column matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart : with(linalg, multiply) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "According to the \+ multiplication conformability rule, the product of a rectangular matri x of order " }{TEXT 275 1 "(" }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 276 3 " \327 " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 277 1 ")" }{TEXT -1 10 " and an " }{TEXT 278 1 "(" }{XPPEDIT 18 0 "m" "6#%\"mG" } {TEXT 279 3 " \327 " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 280 1 ")" } {TEXT -1 2 " " }{TEXT 263 13 "column matrix" }{TEXT -1 52 " is possi ble only if the number of columns in the " }{TEXT 281 1 "(" } {XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 282 3 " \327 " }{XPPEDIT 18 0 "n" " 6#%\"nG" }{TEXT 283 1 ")" }{TEXT -1 48 " matrix is equal to the numbe r of rows in the " }{TEXT 284 1 "(" }{XPPEDIT 18 0 "m" "6#%\"mG" } {TEXT 285 3 " \327 " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 286 1 ")" } {TEXT -1 2 " " }{TEXT 262 13 "column matrix" }{TEXT -1 9 ", i.e. " }{XPPEDIT 18 0 "n=m" "6#/%\"nG%\"mG" }{TEXT -1 29 ". The product matr ix is an " }{TEXT 287 1 "(" }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 288 3 " \327 " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 289 1 ")" }{TEXT -1 2 " " }{TEXT 271 13 "column matrix" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For exampl e, consider a " }{TEXT 290 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 291 3 " \327 " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 292 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 259 1 "A" }{TEXT -1 10 "] given as" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "A := matrix(3, 4, [a[11], a [12], a[13], a[14], a[21], a[22], a[23], a[24], a[31], a[32], a[33], a [34]]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-% 'matrixG6#7%7&&%\"aG6#\"#6&F+6#\"#7&F+6#\"#8&F+6#\"#97&&F+6#\"#@&F+6# \"#A&F+6#\"#B&F+6#\"#C7&&F+6#\"#J&F+6#\"#K&F+6#\"#L&F+6#\"#M" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "and a " }{TEXT 293 1 "(" }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 294 3 " \327 " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 295 1 ")" }{TEXT -1 2 " " }{TEXT 261 13 "column matrix" }{TEXT -1 3 " [" }{TEXT 260 2 "CM" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "CM := matrix(4, 1, [cm[11], cm[21], cm[31], cm[41]]) : CM = \+ matrix(CM) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#CMG-%'matrixG6#7&7# &%#cmG6#\"#67#&F+6#\"#@7#&F+6#\"#J7#&F+6#\"#T" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The product [ " }{TEXT 269 1 "A" }{TEXT -1 3 "] [" }{TEXT 270 2 "CM" }{TEXT -1 21 "] is the following " }{TEXT 296 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 297 5 " \327 1)" }{TEXT -1 16 " column matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "`A CM` := multiply(A, CM) : A*CM \+ = matrix(`A CM`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\"\"%# CMGF&-%'matrixG6#7%7#,**&&%\"aG6#\"#6F&&%#cmGF1F&F&*&&F06#\"#7F&&F46# \"#@F&F&*&&F06#\"#8F&&F46#\"#JF&F&*&&F06#\"#9F&&F46#\"#TF&F&7#,**&&F0F :F&F3F&F&*&&F06#\"#AF&F9F&F&*&&F06#\"#BF&F@F&F&*&&F06#\"#CF&FGF&F&7#,* *&&F0FAF&F3F&F&*&&F06#\"#KF&F9F&F&*&&F06#\"#LF&F@F&F&*&&F06#\"#MF&FGF& F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "This matrix multiplication may be displayed in \"like-in- a-book\" form, namely" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "ma trix(A) * matrix(CM) = matrix(`A CM`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%'matrixG6#7%7&&%\"aG6#\"#6&F+6#\"#7&F+6#\"#8&F+6#\"#97&&F+6 #\"#@&F+6#\"#A&F+6#\"#B&F+6#\"#C7&&F+6#\"#J&F+6#\"#K&F+6#\"#L&F+6#\"#M \"\"\"-F&6#7&7#&%#cmGF,7#&FWF97#&FWFF7#&FW6#\"#TFQ-F&6#7%7#,**&F*FQFVF QFQ*&F.FQFYFQFQ*&F1FQFenFQFQ*&F4FQFgnFQFQ7#,**&F8FQFVFQFQ*&F;FQFYFQFQ* &F>FQFenFQFQ*&FAFQFgnFQFQ7#,**&FEFQFVFQFQ*&FHFQFYFQFQ*&FKFQFenFQFQ*&FN FQFgnFQFQ" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 264 56 "Numerical example for this type of matrix multipl ication" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Let a " }{TEXT 298 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$ " }{TEXT 299 3 " \327 " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 300 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 265 1 "A" }{TEXT -1 9 "] and a " } {TEXT 301 1 "(" }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 302 3 " \327 " } {XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 303 1 ")" }{TEXT -1 17 " column m atrix [" }{TEXT 266 2 "CM" }{TEXT -1 13 "] be given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "A := matrix(3, 4, [1, 2, 1, 0, 0, 1 , 1, 3, 1, 2, 1, 4]) : CM := matrix(4, 1, [2, 1, 0, -1]) :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "A = matrix(A) ; CM = matri x(CM) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7&\"\" \"\"\"#F*\"\"!7&F,F*F*\"\"$7&F*F+F*\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#CMG-%'matrixG6#7&7#\"\"#7#\"\"\"7#\"\"!7#!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "T he product [" }{TEXT 267 1 "A" }{TEXT -1 3 "] [" }{TEXT 268 2 "CM" } {TEXT -1 21 "] is the following " }{TEXT 304 1 "(" }{XPPEDIT 18 0 "3 " "6#\"\"$" }{TEXT 305 5 " \327 1)" }{TEXT -1 16 " column matrix:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "`A CM` := multiply(A, CM) \+ : A*CM = matrix(`A CM`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG \"\"\"%#CMGF&-%'matrixG6#7%7#\"\"%7#!\"#7#\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "or, in \"lik e-in-a-book\" form," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "matr ix(A) * matrix(CM) = matrix(`A CM`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%'matrixG6#7%7&\"\"\"\"\"#F*\"\"!7&F,F*F*\"\"$7&F*F+F*\"\"%F*-F &6#7&7#F+7#F*7#F,7#!\"\"F*-F&6#7%7#F07#!\"#F6" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 272 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 4 "N.B." }{TEXT -1 127 " This specific type of matrix multiplicat ion finds application in solving systems of linear algebraic inhomogen eous equations." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "[ For solving systems of linear algebraic inhomogeneous equations, refer to Unit (19). ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 269 "" 0 "" {TEXT 313 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Proceed to Unit (7) for \"" }{TEXT 312 25 "Special types of mat rices" }{TEXT -1 2 "\"." }}}{EXCHG {PARA 258 "" 0 "" {TEXT 311 67 "--- ----------------------------------------------------------------" }}}} {MARK "24 0 0" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }