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0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 289 1 {CSTYLE " " -1 -1 "" 0 1 4 0 0 0 1 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 290 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 291 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 292 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 293 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 294 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 295 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 296 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 297 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 298 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 299 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 300 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 301 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 302 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 303 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 284 "" 0 "" {TEXT 614 39 "MATRICES AND MATRIX OPE RATIONS: Unit 14" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{PARA 285 "" 0 "" {TEXT 616 23 "Dr. Wlodzislaw Kostecki" }}{PARA 286 "" 0 " " {TEXT -1 53 "The Papua New Guinea University of Technology (PNGUT)" }}{PARA 287 "" 0 "" {TEXT -1 54 "Department of Electrical and Communic ation Engineering" }}{PARA 288 "" 0 "" {TEXT -1 20 "Lae, Morobe Provin ce" }}{PARA 289 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 290 "" 0 "" {TEXT 615 41 "Copyright \251 200 0 by Wlodzislaw Kostecki" }}{PARA 291 "" 0 "" {TEXT 617 19 "All right s reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 292 "" 0 "" {TEXT 618 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 257 4 "(14)" }{TEXT 460 1 " " }{TEXT 459 23 "The inverse of a \+ matrix" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 769 10 "OBJECTIVES" }{TEXT 770 1 ":" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 771 1 "\225" } {TEXT -1 45 " To provide alternative definitions of the " }{TEXT 772 7 "inverse" }{TEXT -1 6 " or " }{TEXT 773 10 "reciprocal" } {TEXT -1 73 " of a square matrix and state the necessary condition fo r its existence." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 774 1 "\225" }{TEXT -1 58 " To provide alternative methods of matrix inversion with " } {TEXT 775 5 "Maple" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 776 1 "\225" }{TEXT -1 66 " To specify and illustrate some properties of the matrix inverse." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 778 1 "\225" }{TEXT -1 63 " To show how the multiplicative inverse matrix may be u sed in " }{TEXT 777 5 "Maple" }{TEXT -1 66 " for computing the \"quoti ent\" of two square non-singular matrices." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "restart : \+ with(linalg, adj, coldim, det, diag, inverse, minor, multiply, rowdim , transpose) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 260 7 "inverse" }{TEXT -1 6 " or " }{TEXT 458 10 "reciprocal" }{TEXT -1 7 " of a " }{TEXT 261 6 "square" }{TEXT -1 32 " matrix is the quotient of the " }{TEXT 258 7 "adjoint" }{TEXT -1 25 " of the matrix and the " }{TEXT 259 11 "de terminant" }{TEXT -1 15 " of the matrix" }}}{EXCHG {PARA 268 "" 0 "" {TEXT 367 3 "Inv" }{TEXT -1 1 "[" }{TEXT 337 1 "A" }{TEXT -1 1 "]" } {TEXT 514 6 " = Adj" }{TEXT -1 1 "[" }{TEXT 338 1 "A" }{TEXT -1 2 "]/ " }{TEXT 368 3 "Det" }{TEXT -1 1 "[" }{TEXT 339 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "The inverse of a matrix exists " }{TEXT 335 2 "if" } {TEXT -1 7 " and " }{TEXT 336 7 "only if" }{TEXT -1 22 " its determ inant is " }{TEXT 19 3 "not" }{TEXT -1 2 " " }{XPPMATH 20 "6#%%zeroG " }{TEXT -1 43 ". A matrix that has an inverse is called " }{TEXT 455 12 "non-singular" }{TEXT -1 6 " or " }{TEXT 456 10 "invertible" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Consider a " }{TEXT 515 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 263 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 516 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 262 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "A := matr ix(2, 2, [a[11], a[12], a[21], a[22]]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$&%\"aG6#\"#6&F+6#\"#77 $&F+6#\"#@&F+6#\"#A" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "The matrix " }{TEXT 265 7 "invers e" }{TEXT -1 17 " to the matrix [" }{TEXT 264 1 "A" }{TEXT -1 8 "] is a " }{TEXT 517 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 518 3 " \+ \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 519 1 ")" }{TEXT -1 79 " m atrix, which can be obtained using any of the following alternative me thods." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 766 8 "Method 1" }{TEXT -1 44 ". Using the defining formula for the matrix " }{TEXT 767 7 "inverse" }{TEXT -1 9 " and the " } {TEXT 768 5 "evalm" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "`inv(A)` := evalm(adj(A)/det(A)) : Inv(A) = ma trix(`inv(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$InvG6#%\"AG-%' matrixG6#7$7$*&&%\"aG6#\"#A\"\"\",&*&&F/6#\"#6F2F.F2F2*&&F/6#\"#7F2&F/ 6#\"#@F2!\"\"F?,$*&F9F2F3F?F?7$,$*&F " 0 "" {MPLTEXT 1 0 54 "`inv(A)` \+ := inverse(A) : Inv(A) = matrix(`inv(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$InvG6#%\"AG-%'matrixG6#7$7$*&&%\"aG6#\"#A\"\"\",&*& &F/6#\"#6F2F.F2F2*&&F/6#\"#7F2&F/6#\"#@F2!\"\"F?,$*&F9F2F3F?F?7$,$*&F< F2F3F?F?*&F5F2F3F?" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 464 8 "Method 3" }{TEXT -1 19 ". Using the power " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 10 " and the " } {TEXT 465 5 "evalm" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "`inv(A)` := evalm(A^(-1)) : Inv(A) = matrix(`i nv(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$InvG6#%\"AG-%'matrixG 6#7$7$*&&%\"aG6#\"#A\"\"\",&*&&F/6#\"#6F2F.F2F2*&&F/6#\"#7F2&F/6#\"#@F 2!\"\"F?,$*&F9F2F3F?F?7$,$*&F " 0 "" {MPLTEXT 1 0 68 " B := matrix(3, 3, [3, -2, 1, -2, 6, 4, 1, 4, 8]) : B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7%7%\"\"$!\"#\"\" \"7%F+\"\"'\"\"%7%F,F/\"\")" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "The inverse of the matrix [" } {TEXT 268 1 "B" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "`inv(B)` := inverse(B) : Inv(B) = matrix(`inv(B)`) \+ ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$InvG6#%\"BG-%'matrixG6#7%7%# \"#;\"#@#\"#5F/#!\"\"\"\"$7%F0#\"#B\"#UF27%F2F2#\"\"\"F4" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "F loating-point evaluation of the inverse of matrix [" }{TEXT 269 1 "B" }{TEXT -1 7 "] gives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "`in v(B)` := evalf(evalm(`inv(B)`)) : Inv(B) = matrix(`inv(B)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$InvG6#%\"BG-%'matrixG6#7%7%$\"+>w/ >w!#5$\"+iZ!>w%F/$!+LLLLLF/7%F0$\"+w/>waF/F27%F2F2$\"+LLLLLF/" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 261 "" 0 "" {TEXT 298 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 470 4 "N.B." }{TEXT -1 65 " Some sources define a matrix inverse using the concept of the " }{TEXT 621 15 "cofactor \+ matrix" }{TEXT -1 119 " \226 refer to Unit (13) for the definition of \+ that matrix. According to this approach, the inverse is defined as fol lows." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 471 7 "inverse" }{TEXT -1 22 " of a squa re matrix [" }{TEXT 474 1 "A" }{TEXT -1 26 "] is the quotient of the \+ " }{TEXT 472 9 "transpose" }{TEXT -1 10 " of the " }{TEXT 473 15 "co factor matrix" }{TEXT -1 19 " associated with [" }{TEXT 477 1 "A" } {TEXT -1 11 "] and the " }{TEXT 475 11 "determinant" }{TEXT -1 6 " o f [" }{TEXT 476 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 281 "" 0 "" {TEXT 481 3 "Inv" }{TEXT -1 1 "[" }{TEXT 478 1 "A" }{TEXT -1 1 "]" } {TEXT 522 18 " = Transp(Cofactor" }{TEXT -1 1 "[" }{TEXT 479 1 "A" } {TEXT -1 1 "]" }{TEXT 483 1 ")" }{TEXT -1 1 "/" }{TEXT 482 3 "Det" } {TEXT -1 1 "[" }{TEXT 480 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 523 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 484 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 524 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 485 1 "A" }{TEXT -1 10 "] given as" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "A := matrix(3, 3, [2, 1, 3, \+ 4, 2, -1, 2, -1, 1]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"#\"\"\"\"\"$7%\"\"%F*!\"\"7%F*F/F+" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "and obtain the inverse of [" }{TEXT 486 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 488 6 "Step 1" }{TEXT -1 34 ". Compute the cofactor matrix of [ " }{TEXT 487 1 "A" }{TEXT -1 19 "] using the double " }{TEXT 489 3 "fo r" }{TEXT -1 16 "-loop construct:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "cofactor(A) := matrix(3, 3) :" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 106 "for i to rowdim(A) do for j to coldim(A) do c ofactor(A)[i,j] := (-1)^(i+j)*det(minor(A, i, j)) : od : od :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "cofactor(A) := matrix(cofact or(A)) : Cofactor(A) = matrix(cofactor(A)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)CofactorG6#%\"AG-%'matrixG6#7%7%\"\"\"!\"'!\")7%!\" %F1\"\"%7%!\"(\"#9\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 490 6 "Step 2" }{TEXT -1 47 ". Compute th e transpose of the cofactor matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`transp(cofactor(A))` := transpose(cofactor(A)) :" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Transp(Cofactor(A)) = matri x(`transp(cofactor(A))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Tran spG6#-%)CofactorG6#%\"AG-%'matrixG6#7%7%\"\"\"!\"%!\"(7%!\"'F1\"#97%! \")\"\"%\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 492 6 "Step 3" }{TEXT -1 30 ". Compute the deter minant of [" }{TEXT 491 1 "A" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : Det(A) = `det(A)` ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG!#G" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 494 6 "Step 4" } {TEXT -1 26 ". Compute the inverse of [" }{TEXT 493 1 "A" }{TEXT -1 40 "] using the definition provided earlier:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "`inv(A)` := evalm(`transp(cofactor(A))`/`det(A)` ) : Inv(A) = matrix(`inv(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%$InvG6#%\"AG-%'matrixG6#7%7%#!\"\"\"#G#\"\"\"\"\"(#F1\"\"%7%#\"\"$\" #9F0#F.\"\"#7%#F:F2#F.F2\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 495 6 "Step 5" }{TEXT -1 31 ". Verif y this result using the " }{TEXT 496 7 "inverse" }{TEXT -1 10 " functi on:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "`inv(A)` := inverse( A) : Inv(A) = matrix(`inv(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%$InvG6#%\"AG-%'matrixG6#7%7%#!\"\"\"#G#\"\"\"\"\"(#F1\"\"%7%#\"\"$ \"#9F0#F.\"\"#7%#F:F2#F.F2\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Both matrix inverses are equ al." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 280 "" 0 "" {TEXT 469 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 671 4 "N.B." }{TEXT -1 10 " If the " } {TEXT 674 11 "commutative" }{TEXT -1 62 " property of multiplication \+ holds for non-singular matrices [" }{TEXT 672 1 "A" }{TEXT -1 7 "] and [" }{TEXT 673 1 "B" }{TEXT -1 58 "], then it also holds for the follo wing pairs of matrices:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "1. \+ " }{TEXT 681 3 "Inv" }{TEXT -1 1 "[" }{TEXT 675 1 "A" }{TEXT -1 2 "] " }{TEXT 682 3 "Inv" }{TEXT -1 1 "[" }{TEXT 678 1 "B" }{TEXT -1 1 "]" }{TEXT 680 6 " = Inv" }{TEXT -1 1 "[" }{TEXT 676 1 "B" }{TEXT -1 2 "] " }{TEXT 679 3 "Inv" }{TEXT -1 1 "[" }{TEXT 677 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "2. \+ " }{TEXT 689 3 "Inv" }{TEXT -1 1 "[" }{TEXT 683 1 "A" }{TEXT -1 3 "] [" }{TEXT 686 1 "B" }{TEXT -1 1 "]" }{TEXT 688 3 " = " }{TEXT -1 1 "[" }{TEXT 684 1 "B " }{TEXT -1 2 "] " }{TEXT 687 3 "Inv" }{TEXT -1 1 "[" }{TEXT 685 1 "A " }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "3. \+ [" }{TEXT 690 1 "A" } {TEXT -1 2 "] " }{TEXT 695 3 "Inv" }{TEXT -1 1 "[" }{TEXT 693 1 "B" } {TEXT -1 1 "]" }{TEXT 694 6 " = Inv" }{TEXT -1 1 "[" }{TEXT 691 1 "B" }{TEXT -1 3 "] [" }{TEXT 692 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Exemplaril y, consider the commuting matrices [" }{TEXT 696 1 "A" }{TEXT -1 7 "] \+ and [" }{TEXT 697 1 "B" }{TEXT -1 33 "] that are used in Unit (4), i.e ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "A := matrix(2, 2, [6, 8, 4, 6]) : B := matrix(2, 2, [15, 20, 10, 15]) : A = matrix(A) \+ ; B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG 6#7$7$\"\"'\"\")7$\"\"%F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%' matrixG6#7$7$\"#:\"#?7$\"#5F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "1(a). The product " }{TEXT 700 3 "Inv" }{TEXT -1 1 "[" }{TEXT 698 1 "A" }{TEXT -1 2 "] " }{TEXT 701 3 "Inv" }{TEXT -1 1 "[" }{TEXT 699 1 "B" }{TEXT -1 5 "] is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "`inv(A) inv(B)` := evalm(inv erse(A) &* inverse(B)) : Inv(A) * Inv(B) = matrix(`inv(A) inv(B)`) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$InvG6#%\"AG\"\"\"-F&6#%\"BGF )-%'matrixG6#7$7$#\"#<\"#5#!#7\"\"&7$#!\"'F7F2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "1(b). The pr oduct " }{TEXT 704 3 "Inv" }{TEXT -1 1 "[" }{TEXT 702 1 "B" }{TEXT -1 2 "] " }{TEXT 705 3 "Inv" }{TEXT -1 1 "[" }{TEXT 703 1 "A" }{TEXT -1 5 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "`inv(B) inv (A)` := evalm(inverse(B) &* inverse(A)) : Inv(B) * `Inv(A)` = matrix (`inv(B) inv(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$InvG6#%\" BG\"\"\"%'Inv(A)GF)-%'matrixG6#7$7$#\"#<\"#5#!#7\"\"&7$#!\"'F5F0" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "The resultant matrices of 1(a) and 1(b) are equal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "2 (a). The product " }{TEXT 708 3 "Inv" }{TEXT -1 1 "[" }{TEXT 706 1 "A " }{TEXT -1 3 "] [" }{TEXT 707 1 "B" }{TEXT -1 5 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "`inv(A) B` := evalm(inverse(A) &* B ) : Inv(A) * B = matrix(`inv(A) B`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$InvG6#%\"AG\"\"\"%\"BGF)-%'matrixG6#7$7$#\"\"&\"\"#\"\"!7$ F3F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 20 "2(b). The product [" }{TEXT 709 1 "B" }{TEXT -1 2 "] \+ " }{TEXT 711 3 "Inv" }{TEXT -1 1 "[" }{TEXT 710 1 "A" }{TEXT -1 5 "] \+ is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "`B inv(A)` := evalm(B &* inverse(A)) : B * `Inv(A)` = matrix(`B inv(A)`) ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/*&%\"BG\"\"\"%'Inv(A)GF&-%'matrixG6#7$7$#\"\"& \"\"#\"\"!7$F0F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "The resultant matrices of 2(a) and 2(b) a re equal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 20 "3(a). The product [" }{TEXT 712 1 "A" }{TEXT -1 2 "] " }{TEXT 714 3 "Inv" }{TEXT -1 1 "[" }{TEXT 713 1 "B" }{TEXT -1 5 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "`A inv(B)` := e valm(A &* inverse(B)) : A * Inv(B) = matrix(`A inv(B)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\"\"-%$InvG6#%\"BGF&-%'matrixG6#7$ 7$#\"\"#\"\"&\"\"!7$F3F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "3(b). The product " }{TEXT 717 3 "Inv" }{TEXT -1 1 "[" }{TEXT 715 1 "B" }{TEXT -1 3 "] [" }{TEXT 716 1 "A" }{TEXT -1 5 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "` inv(B) A` := evalm(inverse(B) &* A) : `Inv(B)` * A = matrix(`inv(B) \+ A`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%'Inv(B)G\"\"\"%\"AGF&-%'m atrixG6#7$7$#\"\"#\"\"&\"\"!7$F0F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "The resultant matrices \+ of 3(a) and 3(b) are equal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 298 "" 0 "" {TEXT 670 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 498 4 "N.B." }{TEXT -1 64 " If two matrices satisfy the commutative law of multiplication " }{TEXT 669 3 "and" }{TEXT -1 46 " their matrix product is the unit \+ matrix, i.e." }}}{EXCHG {PARA 283 "" 0 "" {TEXT -1 1 "[" }{TEXT 499 1 "A" }{TEXT -1 3 "] [" }{TEXT 500 1 "B" }{TEXT -1 1 "]" }{TEXT 525 3 " \+ = " }{TEXT -1 1 "[" }{TEXT 501 1 "B" }{TEXT -1 3 "] [" }{TEXT 502 1 "A " }{TEXT -1 1 "]" }{TEXT 526 3 " = " }{TEXT -1 1 "[" }{TEXT 503 1 "U" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "then matrix [" }{TEXT 504 1 "B" }{TEXT -1 28 "] is the inverse of matrix [" }{TEXT 505 1 "A" }{TEXT -1 2 "]. " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Exemplarily, consider " }{TEXT 527 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 528 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 529 1 ")" }{TEXT -1 12 " matrices [" }{TEXT 506 1 "A" }{TEXT -1 7 "] and [" }{TEXT 507 1 "B" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "A := matrix(2, 2, [3, 4, 2, 3]) : B := matrix(2, 2, [3, -4, -2, 3]) : A = matrix(A) ; B = matrix(B ) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$\"\"$\"\" %7$\"\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7$7$\" \"$!\"%7$!\"#F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(a) The product [" }{TEXT 508 1 "A" } {TEXT -1 3 "] [" }{TEXT 509 1 "B" }{TEXT -1 11 "] is the " }{TEXT 530 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 531 3 " \327 " } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 532 1 ")" }{TEXT -1 14 " unit matr ix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`AB` := multiply(A, \+ B) : `A B` = matrix(`AB`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A~ BG-%'matrixG6#7$7$\"\"\"\"\"!7$F+F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(b) The product [" } {TEXT 510 1 "B" }{TEXT -1 3 "] [" }{TEXT 511 1 "A" }{TEXT -1 11 "] is the " }{TEXT 533 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 534 3 " \+ \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 535 1 ")" }{TEXT -1 14 " u nit matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`BA` := mult iply(B, A) : `B A` = matrix(`BA`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$B~AG-%'matrixG6#7$7$\"\"\"\"\"!7$F+F*" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "(c) The inverse of matrix [" }{TEXT 512 1 "A" }{TEXT -1 26 "] is the following matrix :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "`inv(A)` := inverse(A) : Inv(A) = matrix(`inv(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %$InvG6#%\"AG-%'matrixG6#7$7$\"\"$!\"%7$!\"#F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "which is equ al to matrix [" }{TEXT 513 1 "B" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 282 "" 0 "" {TEXT 497 5 "* * *" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 638 4 "N.B." }{TEXT -1 28 " If inversion of a matrix [" }{TEXT 637 1 "A" }{TEXT -1 34 "] does not change the matrix, then" }}}{EXCHG {PARA 296 "" 0 "" {TEXT 639 1 "(" }{TEXT -1 1 "[" }{TEXT 644 1 "U" } {TEXT -1 1 "]" }{TEXT 640 3 " \226 " }{TEXT -1 1 "[" }{TEXT 645 1 "A" }{TEXT -1 1 "]" }{TEXT 641 3 ") (" }{TEXT -1 1 "[" }{TEXT 646 1 "U" } {TEXT -1 1 "]" }{TEXT 642 3 " + " }{TEXT -1 1 "[" }{TEXT 647 1 "A" } {TEXT -1 1 "]" }{TEXT 643 4 ") = " }{TEXT -1 1 "[" }{TEXT 648 1 "0" } {TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Exemplarily, consider a non-singular " } {TEXT 651 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 650 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 652 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 649 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "A := matrix(3, 3, [1, 0, 0, 0, 0, 1, 0, 1, 0]) : \+ A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6# 7%7%\"\"\"\"\"!F+7%F+F+F*7%F+F*F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "(a) The inverse of [" } {TEXT 653 1 "A" }{TEXT -1 15 "] is the matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "`inv(A)` := inverse(A) : Inv(A) = matrix(`inv( A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$InvG6#%\"AG-%'matrixG6#7 %7%\"\"\"\"\"!F.7%F.F.F-7%F.F-F." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "which is equal to the matr ix [" }{TEXT 654 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "(b) With the appr iopriately sized unit matrix [" }{TEXT 664 1 "U" }{TEXT -1 7 "], i.e. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "U := diag(1, 1, 1) : \+ U = matrix(U) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"UG-%'matrixG6#7 %7%\"\"\"\"\"!F+7%F+F*F+7%F+F+F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "the product " }{TEXT 655 1 "(" }{TEXT -1 1 "[" }{TEXT 660 1 "U" }{TEXT -1 1 "]" }{TEXT 656 3 " \+ \226 " }{TEXT -1 1 "[" }{TEXT 661 1 "A" }{TEXT -1 1 "]" }{TEXT 657 3 " ) (" }{TEXT -1 1 "[" }{TEXT 662 1 "U" }{TEXT -1 1 "]" }{TEXT 658 3 " + " }{TEXT -1 1 "[" }{TEXT 663 1 "A" }{TEXT -1 1 "]" }{TEXT 659 1 ")" } {TEXT -1 4 " is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "`(U-A) \+ (U+A)` := evalm((U-A) &* (U+A)) : '(U - A) * (U + A)' = matrix(`(U-A ) (U+A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"UG\"\"\"%\"AG! \"\"F',&F&F'F(F'F'-%'matrixG6#7%7%\"\"!F0F0F/F/" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "which is a \+ " }{TEXT 666 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 665 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 667 1 ")" }{TEXT -1 2 " " } {XPPEDIT 18 0 "zero" "6#%%zeroG" }{TEXT -1 9 " matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 297 "" 0 "" {TEXT 668 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 4 "N.B." }{TEXT -1 22 " The product of the " }{TEXT 272 7 "inverse" }{TEXT -1 23 " of a matrix and the " }{TEXT 273 6 "m atrix" }{TEXT -1 20 " itself obeys the " }{TEXT 270 15 "commutative \+ law" }{TEXT -1 30 ". The product matrix is the " }{TEXT 271 4 "unit " }{TEXT -1 8 " matrix" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 369 4 "(Inv " }{TEXT -1 1 "[" }{TEXT 275 1 "A" }{TEXT -1 1 "]" }{TEXT 536 1 ")" } {TEXT -1 2 " [" }{TEXT 276 1 "A" }{TEXT -1 1 "]" }{TEXT 538 3 " = " } {TEXT -1 1 "[" }{TEXT 277 1 "A" }{TEXT -1 2 "] " }{TEXT 537 4 "(Inv" } {TEXT -1 1 "[" }{TEXT 278 1 "A" }{TEXT -1 1 "]" }{TEXT 539 4 ") = " } {TEXT -1 1 "[" }{TEXT 279 1 "U" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 540 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 280 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 541 1 ")" }{TEXT -1 21 " invertible matrix [" }{TEXT 281 1 "A" }{TEXT -1 10 "] given a s" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "A := matrix(3, 3, [2, \+ 1, 0, 1, -1, 3, 0, 6, 2]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"#\"\"\"\"\"!7%F+!\"\"\"\"$7%F ,\"\"'F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 17 "(a) The product " }{TEXT 542 4 "(Inv" }{TEXT -1 1 "[" }{TEXT 282 1 "A" }{TEXT -1 1 "]" }{TEXT 543 1 ")" }{TEXT -1 2 " \+ [" }{TEXT 283 1 "A" }{TEXT -1 5 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "`inv(A) A` := multiply(inverse(A), A) : Inv(A)*A = \+ matrix(`inv(A) A`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$InvG6#% \"AG\"\"\"F(F)-%'matrixG6#7%7%F)\"\"!F/7%F/F)F/7%F/F/F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "( b) The product [" }{TEXT 284 1 "A" }{TEXT -1 2 "] " }{TEXT 544 4 "(In v" }{TEXT -1 1 "[" }{TEXT 285 1 "A" }{TEXT -1 1 "]" }{TEXT 545 1 ")" } {TEXT -1 4 " is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "`A inv( A)` := multiply(A, inverse(A)) : A * `Inv(A)` = matrix(`A inv(A)`) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\"\"%'Inv(A)GF&-%'matrix G6#7%7%F&\"\"!F-7%F-F&F-7%F-F-F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Both product matrices are \+ identical and equal to the " }{TEXT 546 1 "(" }{XPPEDIT 18 0 "3" "6# \"\"$" }{TEXT 547 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 548 1 ")" }{TEXT -1 15 " unit matrix [" }{TEXT 331 1 "U" }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "U = matrix(U) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"UG-%'matrixG6#7%7%\"\"\"\"\"!F+7%F+F*F+7 %F+F+F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 295 " " 0 "" {TEXT 636 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 741 4 "N.B." }{TEXT -1 22 " The product \+ of the " }{TEXT 740 7 "inverse" }{TEXT -1 7 " and " }{TEXT 742 9 "t ranspose" }{TEXT -1 8 " of a " }{TEXT 743 9 "symmetric" }{TEXT -1 17 " matrix is the " }{TEXT 744 4 "unit" }{TEXT -1 8 " matrix" }}} {EXCHG {PARA 302 "" 0 "" {TEXT 747 4 "(Inv" }{TEXT -1 1 "[" }{TEXT 745 1 "A" }{TEXT -1 1 "]" }{TEXT 748 1 ")" }{TEXT -1 1 " " }{TEXT 750 7 "(Transp" }{TEXT -1 1 "[" }{TEXT 749 1 "A" }{TEXT -1 1 "]" }{TEXT 751 4 ") = " }{TEXT -1 1 "[" }{TEXT 746 1 "U" }{TEXT -1 1 "]" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 754 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 752 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 755 1 ")" }{TEXT -1 20 " symmetric matrix [" }{TEXT 753 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "A := m atrix(3, 3, [5, 1, 3, 1, 2, -2, 3, -2, 7]) : A = matrix(A) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"&\"\"\"\"\"$ 7%F+\"\"#!\"#7%F,F/\"\"(" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The product " }{TEXT 757 4 "(Inv " }{TEXT -1 1 "[" }{TEXT 756 1 "A" }{TEXT -1 1 "]" }{TEXT 758 1 ")" } {TEXT -1 1 " " }{TEXT 760 7 "(Transp" }{TEXT -1 1 "[" }{TEXT 759 1 "A " }{TEXT -1 1 "]" }{TEXT 761 1 ")" }{TEXT -1 10 " is the " }{TEXT 763 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 762 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 764 1 ")" }{TEXT -1 13 " unit matr ix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "`inv(A) transp(A)` := evalm(inverse(A) &* transpose(A)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Inv(A) * Transp(A) = matrix(`inv(A) transp(A)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$InvG6#%\"AG\"\"\"-%'TranspGF'F)- %'matrixG6#7%7%F)\"\"!F17%F1F)F17%F1F1F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 303 "" 0 "" {TEXT 765 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 291 4 "N.B." }{TEXT -1 7 " The " }{TEXT 292 7 "inverse" }{TEXT -1 10 " of the " }{TEXT 293 14 "matrix inverse" }{TEXT -1 33 " is equa l to the original matrix" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 370 7 "Inv (Inv" }{TEXT -1 1 "[" }{TEXT 294 1 "A" }{TEXT -1 1 "]" }{TEXT 549 4 ") = " }{TEXT -1 1 "[" }{TEXT 295 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exempla rily, consider a " }{TEXT 550 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 297 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 551 1 ")" } {TEXT -1 10 " matrix [" }{TEXT 296 1 "A" }{TEXT -1 10 "] given as" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "A := matrix(2, 2, [a[11], a [12], a[21], a[22]]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$&%\"aG6#\"#6&F+6#\"#77$&F+6#\"#@&F+6#\"#A " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "The inverse of the inverse of the matrix, " }{TEXT 371 7 "Inv(Inv" }{TEXT -1 1 "[" }{TEXT 300 1 "A" }{TEXT -1 1 "]" }{TEXT 552 1 ")" }{TEXT -1 21 ", is the following " }{TEXT 553 1 "(" } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 299 3 " \327 " }{XPPEDIT 18 0 "2" " 6#\"\"#" }{TEXT 554 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "`inv(inv(A))` := inverse(inverse(A)) : Inv( Inv(A)) = matrix(`inv(inv(A))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%$InvG6#-F%6#%\"AG-%'matrixG6#7$7$&%\"aG6#\"#6&F06#\"#77$&F06#\"#@&F0 6#\"#A" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This operation may be displayed in \"like-in-a-book \" form, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Inv(Inv(ma trix(A))) = matrix(`inv(inv(A))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%$InvG6#-F%6#-%'matrixG6#7$7$&%\"aG6#\"#6&F/6#\"#77$&F/6#\"#@&F/6# \"#AF)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 262 " " 0 "" {TEXT 301 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 302 4 "N.B." }{TEXT -1 7 " The " } {TEXT 303 9 "transpose" }{TEXT -1 10 " of the " }{TEXT 304 14 "matri x inverse" }{TEXT -1 19 " is equal to the " }{TEXT 305 7 "inverse" } {TEXT -1 10 " of the " }{TEXT 306 16 "matrix transpose" }}}{EXCHG {PARA 263 "" 0 "" {TEXT 372 10 "Transp(Inv" }{TEXT -1 1 "[" }{TEXT 307 1 "A" }{TEXT -1 1 "]" }{TEXT 555 14 ") = Inv(Transp" }{TEXT -1 1 " [" }{TEXT 308 1 "A" }{TEXT -1 1 "]" }{TEXT 556 1 ")" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exempla rily, consider a " }{TEXT 557 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 310 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 558 1 ")" } {TEXT -1 10 " matrix [" }{TEXT 309 1 "A" }{TEXT -1 10 "] given as" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "A := matrix(2, 2, [a[11], a [12], a[21], a[22]]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$&%\"aG6#\"#6&F+6#\"#77$&F+6#\"#@&F+6#\"#A " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "(a) The transpose of the matrix inverse, " }{TEXT 408 10 "Transp(Inv" }{TEXT -1 1 "[" }{TEXT 312 1 "A" }{TEXT -1 1 "]" } {TEXT 559 1 ")" }{TEXT -1 21 ", is the following " }{TEXT 560 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 311 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 561 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 89 "`transp(inv(A))` := transpose(inverse(A)) : Transp(Inv(A)) = matrix(`transp(inv(A))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6#-%$InvG6#%\"AG-%'matrixG6#7$7$*&&%\"aG6#\" #A\"\"\",&*&&F26#\"#6F5F1F5F5*&&F26#\"#7F5&F26#\"#@F5!\"\"FB,$*&F?F5F6 FBFB7$,$*&F " 0 "" {MPLTEXT 1 0 89 "`inv(transp(A))` : = inverse(transpose(A)) : Inv(Transp(A)) = matrix(`inv(transp(A))`) \+ ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$InvG6#-%'TranspG6#%\"AG-%'mat rixG6#7$7$*&&%\"aG6#\"#A\"\"\",&*&&F26#\"#6F5F1F5F5*&&F26#\"#7F5&F26# \"#@F5!\"\"FB,$*&F?F5F6FBFB7$,$*&F " 0 "" {MPLTEXT 1 0 49 "Transp(Inv(matrix(A))) = Inv (Transp(matrix(A))) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'TranspG6# -%$InvG6#-%'matrixG6#7$7$&%\"aG6#\"#6&F06#\"#77$&F06#\"#@&F06#\"#A-F(6 #-F%F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 264 " " 0 "" {TEXT 315 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 316 4 "N.B." }{TEXT -1 113 " The inverse of the product of two matrices is the product of the two inversed mat rices but in the reverse order" }}}{EXCHG {PARA 265 "" 0 "" {TEXT 374 4 "Inv(" }{TEXT -1 1 "[" }{TEXT 317 1 "A" }{TEXT -1 3 "] [" }{TEXT 318 1 "B" }{TEXT -1 1 "]" }{TEXT 565 7 ") = Inv" }{TEXT -1 1 "[" } {TEXT 319 1 "B" }{TEXT -1 2 "] " }{TEXT 375 3 "Inv" }{TEXT -1 1 "[" } {TEXT 320 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Exemplarily, consider " } {TEXT 566 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 567 3 " \327 " } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 568 1 ")" }{TEXT -1 12 " matrices \+ [" }{TEXT 321 1 "A" }{TEXT -1 7 "] and [" }{TEXT 322 1 "B" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "A := mat rix(2, 2, [a[11], a[12], a[21], a[22]]) : B := matrix(2, 2, [b[11], \+ b[12], b[21], b[22]]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " A = matrix(A) ; B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %\"AG-%'matrixG6#7$7$&%\"aG6#\"#6&F+6#\"#77$&F+6#\"#@&F+6#\"#A" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7$7$&%\"bG6#\"#6&F+6 #\"#77$&F+6#\"#@&F+6#\"#A" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "(a) The inverse of the product of \+ the two matrices, " }{TEXT 376 4 "Inv(" }{TEXT -1 1 "[" }{TEXT 323 1 "A" }{TEXT -1 3 "] [" }{TEXT 324 1 "B" }{TEXT -1 1 "]" }{TEXT 569 1 ") " }{TEXT -1 21 ", is the following " }{TEXT 570 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 571 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 572 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "`inv(AB)` := inverse(multiply(A, B)) : Inv(A * B) = matrix(`inv(AB)`) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$InvG6#*&% \"AG\"\"\"%\"BGF)-%'matrixG6#7$7$*&,&*&&%\"aG6#\"#@F)&%\"bG6#\"#7F)F)* &&F46#\"#AF)&F8F=F)F)F),***&F46#\"#6F)&F8FCF)F " 0 "" {MPLTEXT 1 0 99 "`inv(B) inv(A)` := multipl y(inverse(B), inverse(A)) : Inv(B) * Inv(A) = matrix(`inv(B) inv(A)` ) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*&-%$InvG6#%\"BG\"\"\"-F&6#%\" AGF)-%'matrixG6#7$7$,&*&*&&%\"bG6#\"#AF)&%\"aGF7F)F)*&,&*&&F66#\"#6F)F 5F)F)*&&F66#\"#7F)&F66#\"#@F)!\"\"F),&*&&F:F?F)F9F)F)*&&F:FCF)&F:FFF)F HF)FHF)*&*&FBF)FNF)F)*&FF)FNF)F)* &FF)FKF)F)*&F" }{TEXT 577 2 " )" }{TEXT -1 12 ", in wh ich " }{TEXT 352 1 "x" }{TEXT -1 7 " is a " }{TEXT 354 14 "local vari able" }{TEXT -1 50 ". In this way, the procedure is applied to each \+ " }{TEXT 356 7 "operand" }{TEXT -1 27 " (matrix element) of the " } {TEXT 357 15 "second argument" }{TEXT -1 22 " (the matrix) of the " } {TEXT 355 3 "map" }{TEXT -1 10 " function." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 65 "`inv(B) inv(A)` := map(x->normal(x, expanded), `inv (B) inv(A)`) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Inv(B) * \+ Inv(A) = matrix(`inv(B) inv(A)`) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# /*&-%$InvG6#%\"BG\"\"\"-F&6#%\"AGF)-%'matrixG6#7$7$*&,&*&&%\"aG6#\"#@F )&%\"bG6#\"#7F)F)*&&F66#\"#AF)&F:F?F)F)F),***&F66#\"#6F)&F:FEF)F>F)FAF )F)**&F6F;F)&F:F7F)F5F)F9F)F)**FDF)F9F)F>F)FJF)!\"\"**FIF)FAF)F5F)FGF) FLFL*&,&*&FIF)FAF)FL*&FDF)F9F)FLF)FBFL7$*&,&*&F>F)FJF)FL*&F5F)FGF)FLF) FBFL*&,&*&FDF)FGF)F)*&FIF)FJF)F)F)FBFL" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The product matri x " }{TEXT 387 3 "Inv" }{TEXT -1 1 "[" }{TEXT 385 1 "B" }{TEXT -1 2 " ] " }{TEXT 388 3 "Inv" }{TEXT -1 1 "[" }{TEXT 386 1 "A" }{TEXT -1 65 " ] displayed in the compact form is clearly equal to the matrix " } {TEXT 391 4 "Inv(" }{TEXT -1 1 "[" }{TEXT 389 1 "A" }{TEXT -1 3 "] [" }{TEXT 390 1 "B" }{TEXT -1 1 "]" }{TEXT 578 1 ")" }{TEXT -1 8 " of (a) ." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 266 "" 0 " " {TEXT 327 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 286 4 "N.B." }{TEXT -1 109 " The determi nant of a non-singular square matrix is the reciprocal of the determin ant of the inversed matrix" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 379 3 "D et" }{TEXT -1 1 "[" }{TEXT 287 1 "A" }{TEXT -1 1 "]" }{TEXT 579 3 " = \+ " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT -1 1 "/" }{TEXT 380 7 "Det(Inv " }{TEXT -1 1 "[" }{TEXT 288 1 "A" }{TEXT -1 1 "]" }{TEXT 580 1 ")" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 581 7 "(2 \327 2)" }{TEXT -1 10 " matrix [" }{TEXT 289 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A := matrix(2, 2, [1, 3, 2, 4]) : \+ A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6# 7$7$\"\"\"\"\"$7$\"\"#\"\"%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "The inverse of [" }{TEXT 290 1 " A" }{TEXT -1 20 "] is the following " }{TEXT 582 7 "(2 \327 2)" } {TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "`i nv(A)` := inverse(A) : Inv(A) = matrix(`inv(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$InvG6#%\"AG-%'matrixG6#7$7$!\"##\"\"$\"\"#7$\"\" \"#!\"\"F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "(a) The determinant of the matrix, " }{TEXT 381 3 "Det" }{TEXT -1 1 "[" }{TEXT 329 1 "A" }{TEXT -1 6 "], is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : Det(A ) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG!\"# " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "(b) The reciprocal of the determinant of the inversed mat rix, " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT -1 1 "/" }{TEXT 382 7 "De t(Inv" }{TEXT -1 1 "[" }{TEXT 330 1 "A" }{TEXT -1 1 "]" }{TEXT 583 1 " )" }{TEXT -1 5 ", is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "`1 /det(inv(A))` := 1/det(`inv(A)`) : 1/Det(Inv(A)) = `1/det(inv(A))` ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&\"\"\"F%-%$DetG6#-%$InvG6#%\"AG !\"\"!\"#" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 584 4 "N.B." }{TEXT -1 119 " This property of the determ inant is used in solving systems of linear algebraic inhomogeneous equ ations by means of " }{TEXT 461 7 "Cramer\222" }{TEXT -1 29 "s rule \+ \226 refer to Unit (19)." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 267 "" 0 "" {TEXT 328 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 623 4 "N.B." }{TEXT -1 115 " The determinant of the inverse of a non-singular square matr ix is the reciprocal of the determinant of the matrix" }}}{EXCHG {PARA 294 "" 0 "" {TEXT 627 7 "Det(Inv" }{TEXT -1 1 "[" }{TEXT 626 1 " A" }{TEXT -1 1 "]" }{TEXT 628 4 ") = " }{XPPEDIT 18 0 "1" "6#\"\"\"" } {TEXT -1 1 "/" }{TEXT 625 3 "Det" }{TEXT -1 1 "[" }{TEXT 624 1 "A" } {TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Exemplarily, consider the same " }{TEXT 630 7 "(2 \327 2)" }{TEXT -1 10 " matrix [" }{TEXT 629 1 "A" }{TEXT -1 11 "] as above." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "(a) The determinant of the matrix inverse , " }{TEXT 632 7 "Det(Inv" }{TEXT -1 1 "[" }{TEXT 631 1 "A" }{TEXT -1 1 "]" }{TEXT 633 1 ")" }{TEXT -1 5 ", is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "`det(inv(A))` := det(inverse(A)) : Det(Inv(A)) = `det(inv(A))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#-%$Inv G6#%\"AG#!\"\"\"\"#" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "(b) The reciprocal of the determin ant of the matrix, " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT -1 1 "/" } {TEXT 635 3 "Det" }{TEXT -1 1 "[" }{TEXT 634 1 "A" }{TEXT -1 6 "], is " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`1/det(A)` := 1/det(A) \+ : 1/Det(A) = `1/det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&\"\" \"F%-%$DetG6#%\"AG!\"\"#F*\"\"#" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 293 "" 0 "" {TEXT 622 5 "* * *" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 718 4 "N.B." } {TEXT -1 6 " If [" }{TEXT 724 1 "A" }{TEXT -1 7 "] and [" }{TEXT 725 1 "B" }{TEXT -1 51 "] are non-singular matrices of the same order, the n" }}}{EXCHG {PARA 299 "" 0 "" {TEXT 722 7 "Det(Inv" }{TEXT -1 1 "[" } {TEXT 726 1 "A" }{TEXT -1 3 "] [" }{TEXT 719 1 "B" }{TEXT -1 3 "] [" } {TEXT 720 1 "A" }{TEXT -1 1 "]" }{TEXT 723 7 ") = Det" }{TEXT -1 1 "[ " }{TEXT 721 1 "B" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Exemplarily, consider " } {TEXT 729 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 727 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 730 1 ")" }{TEXT -1 25 " non-singu lar matrices [" }{TEXT 728 1 "A" }{TEXT -1 7 "] and [" }{TEXT 731 1 "B " }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "A := matrix(3, 3, [2, 3, 1, 3, -1, -2, -1, 2, 1]) : B := matrix (3, 3, [12, 8, -1, 6, 4, 0, -7, -2, 1]) :" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "A = matrix(A) ; B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"#\"\"$\"\"\"7%F+!\"\"!\"#7 %F.F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7%7%\"#7 \"\")!\"\"7%\"\"'\"\"%\"\"!7%!\"(!\"#\"\"\"" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "(a) The determi nant " }{TEXT 734 7 "Det(Inv" }{TEXT -1 1 "[" }{TEXT 736 1 "A" } {TEXT -1 3 "] [" }{TEXT 732 1 "B" }{TEXT -1 3 "] [" }{TEXT 733 1 "A" } {TEXT -1 1 "]" }{TEXT 735 1 ")" }{TEXT -1 4 " is" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 95 "`det(inv(A) B A)` := det(evalm(inverse(A) &* B &* A)) : Det(Inv(A)*B*A) = `det(inv(A) B A)` ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$DetG6#*(-%$InvG6#%\"AG\"\"\"%\"BGF,F+F,!#;" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "(b) The determinant of [" }{TEXT 737 1 "B" }{TEXT -1 4 "] is" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(B)` := det(B) : Det (B) = `det(B)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"BG!#; " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 300 "" 0 "" {TEXT 738 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 342 4 "N.B." }{TEXT -1 29 " Although the operat ion of " }{TEXT 340 8 "division" }{TEXT -1 5 " is " }{TEXT 341 3 "no t" }{TEXT -1 43 " defined for matrices, the multiplicative " }{TEXT 347 14 "inverse matrix" }{TEXT -1 45 " is used in computing the \"quo tient\" of two " }{TEXT 365 6 "square" }{TEXT -1 1 " " }{TEXT 366 12 " non-singular" }{TEXT -1 26 " matrices. In this sense, " }{TEXT 348 5 " Maple" }{TEXT -1 84 " recognises \"division\" of two matrices and perf orms it according to the relationship" }}}{EXCHG {PARA 269 "" 0 "" {TEXT -1 1 "[" }{TEXT 343 1 "A" }{TEXT -1 3 "]/[" }{TEXT 344 1 "B" } {TEXT -1 1 "]" }{TEXT 585 3 " = " }{TEXT -1 1 "[" }{TEXT 345 1 "A" } {TEXT -1 2 "] " }{TEXT 383 3 "Inv" }{TEXT -1 1 "[" }{TEXT 346 1 "B" } {TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Exemplarily, consider non-singular " } {TEXT 586 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 350 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 587 1 ")" }{TEXT -1 12 " matrices \+ [" }{TEXT 349 1 "A" }{TEXT -1 7 "] and [" }{TEXT 351 1 "B" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "A := mat rix(3, 3, [2, 1, 0, 1, -1, 3, 0, 6, 2]) : B := matrix(3, 3, [3, -2, \+ 1, 0, 4, -3, 2, 5, -1]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "A = matrix(A) ; B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"#\"\"\"\"\"!7%F+!\"\"\"\"$7%F,\"\"'F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7%7%\"\"$!\"#\"\"\"7 %\"\"!\"\"%!\"$7%\"\"#\"\"&!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "(a) The \"quotient\" of th e matrices, [" }{TEXT 359 1 "A" }{TEXT -1 3 "]/[" }{TEXT 360 1 "B" } {TEXT -1 19 "], is returned by " }{TEXT 361 5 "Maple" }{TEXT -1 19 " \+ as the following " }{TEXT 588 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 358 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 589 1 ")" } {TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "`A /B` := evalm(A/B) : A/B = matrix(`A/B`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\"\"%\"BG!\"\"-%'matrixG6#7%7%#\"#;\"#P#F&F0# \"#8F07%#!\"(F0#!#\\F0#\"#HF07%#!#_F0#!#oF0#\"#yF0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "(b) The p roduct matrix [" }{TEXT 362 1 "A" }{TEXT -1 2 "] " }{TEXT 384 3 "Inv " }{TEXT -1 1 "[" }{TEXT 363 1 "B" }{TEXT -1 21 "] is the following \+ " }{TEXT 590 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 364 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 591 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "`A inv(B)` := multiply(A, inverse(B)) : A * Inv(B) = matrix(`A inv(B)`) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*&%\"AG\"\"\"-%$InvG6#%\"BGF&-%'matrixG6#7%7%#\"#;\" #P#F&F2#\"#8F27%#!\"(F2#!#\\F2#\"#HF27%#!#_F2#!#oF2#\"#yF2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "B oth matrices of (a) and (b) are equal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 274 "" 0 "" {TEXT 409 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 410 4 "N.B." }{TEXT -1 35 " It follows from the relationships" }}} {EXCHG {PARA 276 "" 0 "" {TEXT -1 1 "[" }{TEXT 411 1 "A" }{TEXT -1 3 " ]/[" }{TEXT 412 1 "B" }{TEXT -1 1 "]" }{TEXT 592 3 " = " }{TEXT -1 1 " [" }{TEXT 413 1 "A" }{TEXT -1 2 "] " }{TEXT 415 3 "Inv" }{TEXT -1 1 "[ " }{TEXT 414 1 "B" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "and" }}}{EXCHG {PARA 277 " " 0 "" {TEXT 418 7 "Inv(Inv" }{TEXT -1 1 "[" }{TEXT 416 1 "A" }{TEXT -1 1 "]" }{TEXT 593 4 ") = " }{TEXT -1 1 "[" }{TEXT 417 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "that for " }{TEXT 451 6 "square" }{TEXT -1 1 " " } {TEXT 452 12 "non-singular" }{TEXT -1 9 " matrices" }}}{EXCHG {PARA 275 "" 0 "" {TEXT 433 3 "Inv" }{TEXT -1 1 "[" }{TEXT 419 1 "A" }{TEXT -1 2 "]/" }{TEXT 434 3 "Inv" }{TEXT -1 1 "[" }{TEXT 420 1 "K" }{TEXT -1 1 "]" }{TEXT 594 6 " = Inv" }{TEXT -1 1 "[" }{TEXT 421 1 "A" } {TEXT -1 2 "] " }{TEXT 595 8 "\{Inv(Inv" }{TEXT -1 1 "[" }{TEXT 422 1 "K" }{TEXT -1 1 "]" }{TEXT 596 9 ")\} = (Inv" }{TEXT -1 1 "[" }{TEXT 423 1 "A" }{TEXT -1 1 "]" }{TEXT 597 1 ")" }{TEXT -1 2 " [" }{TEXT 424 1 "K" }{TEXT -1 2 "] " }{TEXT 440 1 "=" }{TEXT -1 2 " [" }{TEXT 425 1 "X" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "where the matrix [" }{TEXT 426 1 " X" }{TEXT -1 24 "] satisfies the equation" }}}{EXCHG {PARA 278 "" 0 " " {TEXT -1 1 "[" }{TEXT 427 1 "A" }{TEXT -1 3 "] [" }{TEXT 428 1 "X" } {TEXT -1 1 "]" }{TEXT 598 3 " = " }{TEXT -1 1 "[" }{TEXT 429 1 "K" } {TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "To verify this conclusion, consider the f ollowing " }{TEXT 600 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 599 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 601 1 ")" }{TEXT -1 11 " matrices:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "A := matri x(3, 3, [2, 1, 0, 1, -1, 3, 0, 6, 2]) : X := matrix(3, 3, [3, -2, 1, 0, 4, -3, 2, 5, -1]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "K := matrix(3, 3, [6, 0, -1, 9, 9, 1, 4, 34, -20]) : A = matrix(A) \+ ; X = matrix(X) ; K = matrix(K) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"#\"\"\"\"\"!7%F+!\"\"\"\"$7%F,\"\"'F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"XG-%'matrixG6#7%7%\"\"$!\"#\"\"\"7 %\"\"!\"\"%!\"$7%\"\"#\"\"&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% \"KG-%'matrixG6#7%7%\"\"'\"\"!!\"\"7%\"\"*F.\"\"\"7%\"\"%\"#M!#?" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "(a) The \"quotient\" of the matrices " }{TEXT 437 3 "Inv" } {TEXT -1 1 "[" }{TEXT 435 1 "A" }{TEXT -1 2 "]/" }{TEXT 438 3 "Inv" } {TEXT -1 1 "[" }{TEXT 436 1 "K" }{TEXT -1 21 "] is the following " } {TEXT 602 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 439 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 603 1 ")" }{TEXT -1 8 " matrix" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "`inv(A)/inv(K)` := evalm(in verse(A)/inverse(K)) : Inv(A)/Inv(K) = matrix(`inv(A)/inv(K)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$InvG6#%\"AG\"\"\"-F&6#%\"KG!\"\" -%'matrixG6#7%7%\"\"$!\"#F)7%\"\"!\"\"%!\"$7%\"\"#\"\"&F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "w hich is equal to the matrix [" }{TEXT 441 1 "X" }{TEXT -1 2 "]." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "(b) The product matrix [" }{TEXT 431 1 "A" }{TEXT -1 3 "] [" } {TEXT 432 1 "X" }{TEXT -1 21 "] is the following " }{TEXT 604 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 430 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 605 1 ")" }{TEXT -1 8 " matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`AX` := multiply(A, X) : A * X = matrix(`AX `) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\"\"%\"XGF&-%'matrix G6#7%7%\"\"'\"\"!!\"\"7%\"\"*F1F&7%\"\"%\"#M!#?" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "which is equ al to the matrix [" }{TEXT 442 1 "K" }{TEXT -1 2 "]." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 606 4 "N.B." } {TEXT -1 6 " If [" }{TEXT 448 1 "A" }{TEXT -1 39 "] is a square non-s ingular matrix and [" }{TEXT 449 1 "K" }{TEXT -1 11 "] is a non-" } {XPPMATH 20 "6#%%zeroG" }{TEXT -1 2 " " }{TEXT 454 13 "column matrix " }{TEXT -1 51 " with the number of rows equal to that in matrix [" } {TEXT 450 1 "A" }{TEXT -1 22 "], the relationship [" }{TEXT 445 1 "A " }{TEXT -1 3 "] [" }{TEXT 446 1 "X" }{TEXT -1 1 "]" }{TEXT 607 3 " = \+ " }{TEXT -1 1 "[" }{TEXT 447 1 "K" }{TEXT -1 82 "] is used for repres enting a system of linear algebraic inhomogeneous equations " }{TEXT 453 18 "in the matrix form" }{TEXT -1 24 " \226 refer to Unit (19). " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 279 "" 0 "" {TEXT 443 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 444 4 "N.B." }{TEXT -1 35 " It follows from the relationships" }}}{EXCHG {PARA 272 "" 0 "" {TEXT -1 1 "[" }{TEXT 392 1 "A" }{TEXT -1 3 "]/[" }{TEXT 393 1 "B" }{TEXT -1 1 "]" }{TEXT 608 3 " = " }{TEXT -1 1 "[" }{TEXT 394 1 "A" }{TEXT -1 2 "] " }{TEXT 396 3 " Inv" }{TEXT -1 1 "[" }{TEXT 395 1 "B" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "and" }}} {EXCHG {PARA 273 "" 0 "" {TEXT -1 1 "[" }{TEXT 397 1 "A" }{TEXT -1 2 " ] " }{TEXT 609 4 "(Inv" }{TEXT -1 1 "[" }{TEXT 398 1 "A" }{TEXT -1 1 " ]" }{TEXT 610 4 ") = " }{TEXT -1 1 "[" }{TEXT 399 1 "U" }{TEXT -1 1 "] " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "that for a square non-singular matrix [" }{TEXT 400 1 "A " }{TEXT -1 1 "]" }}}{EXCHG {PARA 271 "" 0 "" {TEXT -1 1 "[" }{TEXT 401 1 "A" }{TEXT -1 3 "]/[" }{TEXT 402 1 "A" }{TEXT -1 1 "]" }{TEXT 611 3 " = " }{TEXT -1 1 "[" }{TEXT 403 1 "U" }{TEXT -1 1 "]" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "To verify this conclusion, consider the above " }{TEXT 612 1 " (" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 405 3 " \327 " }{XPPEDIT 18 0 " 3" "6#\"\"$" }{TEXT 613 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 404 1 " A" }{TEXT -1 29 "] and compute the result of [" }{TEXT 406 1 "A" } {TEXT -1 3 "]/[" }{TEXT 407 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 56 "`A/A` := multiply(A, A^(-1)) : `A/A` = mat rix(`A/A`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A/AG-%'matrixG6#7%7 %\"\"\"\"\"!F+7%F+F*F+7%F+F+F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "or, using the " }{TEXT 457 5 "evalm" }{TEXT -1 10 " function," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "`A/A` := evalm(A &* evalm(A^(-1))) : `A/A` = matrix (`A/A`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A/AG-%'matrixG6#7%7%\" \"\"\"\"!F+7%F+F*F+7%F+F+F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 301 "" 0 "" {TEXT 739 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Proceed to Uni t (15) for \"" }{TEXT 620 34 "Integer exponentiation of matrices" } {TEXT -1 2 "\"." }}}{EXCHG {PARA 270 "" 0 "" {TEXT 619 67 "----------- --------------------------------------------------------" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }