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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 "" 0 288 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 "" 0 289 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 "" 0 290 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 }} {SECT 0 {EXCHG {PARA 277 "" 0 "" {TEXT 511 39 "MATRICES AND MATRIX OPE RATIONS: Unit 11" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{PARA 278 "" 0 "" {TEXT 513 23 "Dr. Wlodzislaw Kostecki" }}{PARA 279 "" 0 " " {TEXT -1 53 "The Papua New Guinea University of Technology (PNGUT)" }}{PARA 280 "" 0 "" {TEXT -1 54 "Department of Electrical and Communic ation Engineering" }}{PARA 281 "" 0 "" {TEXT -1 20 "Lae, Morobe Provin ce" }}{PARA 282 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 283 "" 0 "" {TEXT 512 41 "Copyright \251 200 0 by Wlodzislaw Kostecki" }}{PARA 284 "" 0 "" {TEXT 514 19 "All right s reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 285 "" 0 "" {TEXT 515 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 257 4 "(11)" }{TEXT 406 1 " " }{TEXT 405 27 "The determinant o f a matrix" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 568 10 "OBJECTIVES" }{TEXT 569 1 ":" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 570 1 "\225" } {TEXT -1 17 " To define the " }{TEXT 571 11 "determinant" }{TEXT -1 55 " of a square matrix and introduce the concept of the " }{TEXT 579 5 "order" }{TEXT -1 17 " of determinant." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 580 1 "\225" }{TEXT -1 105 " To provide examples of computat ion of the determinant of matrices with symbolic and numerical element s." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 581 1 "\225" }{TEXT -1 81 " To in vestigate the value of the determinant of some specific types of matri ces." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 582 1 "\225" }{TEXT -1 138 " To investigate properties of determinants of matrices having specific va lues of, or specific relations between, row and column elements." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 583 1 "\225" }{TEXT -1 109 " To investig ate how some operations performed on entire matrices or their componen ts affect the determinant." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 585 1 " \225" }{TEXT -1 42 " To introduce further functions from the " } {TEXT 584 6 "linalg" }{TEXT -1 51 " package and show their use in matr ix computations." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "restart : with(linalg, addcol, a ddrow, coldim, det, diag, mulcol, mulrow, multiply, rowdim, swapcol, s waprow, transpose, vandermonde) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 259 11 "deter minant" }{TEXT -1 25 " of a matrix is defined " }{TEXT 262 4 "only" } {TEXT -1 6 " for " }{TEXT 263 15 "square matrices" }{TEXT -1 85 " an d it is the sum of certain products of the matrix elements. It is, the refore, a " }{TEXT 264 6 "scalar" }{TEXT -1 86 " (real number, compl ex number, or function, depending on the elements of the matrix)." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Exemplarily, consider a square " }{TEXT 449 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 258 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 450 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 260 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "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 $&F+6#\"#@&F+6#\"#A" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 288 11 "determinant" }{TEXT -1 46 " of this matrix is defined as the following " }{TEXT 289 15 "sum of products" }{TEXT -1 27 " of the diagonal elements:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : Det( A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG,&*& &%\"aG6#\"#6\"\"\"&F+6#\"#AF.F.*&&F+6#\"#7F.&F+6#\"#@F.!\"\"" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "This sum is also the " }{TEXT 566 5 "value" }{TEXT -1 25 " of the determinant of [" }{TEXT 565 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 290 "" 0 "" {TEXT 557 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 559 4 "N.B." }{TEXT -1 72 " In textbooks, a common method \+ of visualization of the determinant of [" }{TEXT 562 1 "A" }{TEXT -1 19 "] uses the symbol " }{XPPEDIT 18 0 "abs(A)" "6#-%$absG6#%\"AG" } {TEXT 561 1 "," }{TEXT -1 28 " which is followed by the " }{TEXT 560 1 "=" }{TEXT -1 177 " sign and a square array of elements arrange d exactly as they are displayed above, but enclosed in a pair of verti cal lines. The number of rows or columns of the array is the " } {TEXT 563 5 "order" }{TEXT -1 21 " of the determinant." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "I n the case of the above matrix [" }{TEXT 564 1 "A" }{TEXT -1 38 "], it s determinant is of second order." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 567 5 "Maple" }{TEXT -1 62 " does not offer this type of visualization of the determinant." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 291 "" 0 "" {TEXT 558 5 "* * *" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "As a more advanced example, consider a " }{TEXT 451 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 452 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 453 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 261 1 "B" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "B := matrix(3, 3, [b[11], b[12], b[13], b[21], b[ 22], b[23], b[31], b[32], b[33]]) : B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7%7%&%\"bG6#\"#6&F+6#\"#7&F+6#\" #87%&F+6#\"#@&F+6#\"#A&F+6#\"#B7%&F+6#\"#J&F+6#\"#K&F+6#\"#L" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "The determinant of this matrix is the third-order determinant. Its value is the following sum of products:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(B)` := det(B) : Det(B) = `det(B)` ;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"BG,.*(&%\"bG6#\"#6\"\"\"& F+6#\"#AF.&F+6#\"#LF.F.*(F*F.&F+6#\"#BF.&F+6#\"#KF.!\"\"*(&F+6#\"#@F.& F+6#\"#7F.F2F.F<*(F>F.&F+6#\"#8F.F9F.F.*(&F+6#\"#JF.FAF.F6F.F.*(FIF.FE F.F/F.F<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 282 17 "Numerical example" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Compute the deter minant of a " }{TEXT 454 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 283 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 455 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 284 1 "C" }{TEXT -1 10 "] given as" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "C := matrix(3, 3, [7, 18, 8, 1, 5, 7, 3, 9, 4]) : C = matrix(C) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"CG-%'matrixG6#7%7%\"\"(\"#=\"\")7%\"\"\"\"\"&F*7%\"\"$\"\"* \"\"%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The determinant of [" }{TEXT 290 1 "C" }{TEXT -1 4 "] \+ is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(C)` := det(C) : Det(C) = `det(C)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\" CG!#V" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 280 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 328 4 "N.B." }{TEXT -1 24 " The determin ant of a " }{TEXT 456 1 "(" }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 457 3 " \327 " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 458 1 ")" }{TEXT -1 31 " matrix is the element itself." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 459 1 "(" }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 460 3 " \327 " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 461 1 ")" }{TEXT -1 10 " matr ix [" }{TEXT 329 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 42 "A := matrix(1, 1, [a]) : A = matrix(A) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7#7#%\"aG" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : Det(A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG%\"aG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 268 "" 0 "" {TEXT 330 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 546 4 "N.B." }{TEXT -1 24 " The determinant of a " }{TEXT 556 4 "uni t" }{TEXT -1 13 " matrix is " }{XPPMATH 20 "6#%&unityG" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 549 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 550 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 551 1 ")" }{TEXT -1 15 " unit matrix [" }{TEXT 548 1 "U" } {TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "U := diag (1, 1, 1) : U = matrix(U) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"U G-%'matrixG6#7%7%\"\"\"\"\"!F+7%F+F*F+7%F+F+F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "The determin ant of this matrix is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`d et(U)` := det(U) : Det(U) = `det(U)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"UG\"\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 289 "" 0 "" {TEXT 547 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 323 4 "N.B." }{TEXT -1 6 " If " }{TEXT 324 3 "all" }{TEXT -1 58 " the elements of a row (or of a column) in a matrix are " }{XPPMATH 20 "6#%&zerosG" }{TEXT -1 42 ", then the determinant of the matrix is " }{XPPMATH 20 "6#%%z eroG" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 462 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 463 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 464 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 327 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "A := matrix(3, 3, [a[11], a[12], a[13], 0, 0, 0, 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%\"\"!F5F57 %&F+6#\"#J&F+6#\"#K&F+6#\"#L" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "The determinant of this matrix \+ is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : Det(A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\" AG\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "and the matrix [" }{TEXT 326 1 "A" }{TEXT -1 17 "] i s said to be " }{TEXT 269 8 "singular" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 325 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 389 4 "N.B." }{TEXT -1 6 " If " }{TEXT 390 3 "two" } {TEXT -1 31 " rows of a square matrix are " }{TEXT 391 5 "equal" } {TEXT -1 28 ", then its determinant is " }{XPPMATH 20 "6#%%zeroG" } {TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 465 1 " (" }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 466 3 " \327 " }{XPPEDIT 18 0 " 4" "6#\"\"%" }{TEXT 467 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 392 1 " A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "A := matrix(4, 4, [2, 2, 3, 3, 2, 3, 3, 2, 5, 3, 7, 9, 2, 2, 3, 3] ) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matr ixG6#7&7&\"\"#F*\"\"$F+7&F*F+F+F*7&\"\"&F+\"\"(\"\"*F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "i n which the rows " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT -1 12 " are equal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "T he determinant of this matrix is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : Det(A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 261 "" 0 "" {TEXT 281 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 394 4 "N.B." }{TEXT -1 31 " The determinant of a matrix [" }{TEXT 518 1 "A" }{TEXT -1 103 "] raised to any positive or negative integer \+ power is equal to the determinant raised to the same power" }}}{EXCHG {PARA 272 "" 0 "" {TEXT 397 4 "Det(" }{TEXT -1 1 "[" }{TEXT 395 1 "A" }{TEXT -1 1 "]" }{TEXT 398 1 "^" }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 468 8 ") = (Det" }{TEXT -1 1 "[" }{TEXT 396 1 "A" }{TEXT -1 1 "]" } {TEXT 469 2 ")^" }{XPPEDIT 18 0 "n" "6#%\"nG" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, co nsider a " }{TEXT 470 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 471 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 472 1 ")" }{TEXT -1 23 " non-singular matrix [" }{TEXT 399 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%\"\"#\"\"\"\"\"$7%\"\"%F*!\"\"7%F *F/F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : Det(A) = `det(A)` ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG!#G" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Let the p ositive exponent be " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT -1 29 " an d the negative exponent " }{XPPEDIT 18 0 "-2" "6#,$\"\"#!\"\"" } {TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "(a) The values of " }{TEXT 401 4 "Det(" }{TEXT -1 1 "[" }{TEXT 400 1 "A" }{TEXT -1 1 "]" }{TEXT 402 1 "^" } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 519 1 ")" }{TEXT -1 7 " and " } {TEXT 521 4 "Det(" }{TEXT -1 1 "[" }{TEXT 520 1 "A" }{TEXT -1 1 "]" } {TEXT 522 2 "^(" }{XPPEDIT 18 0 "-2" "6#,$\"\"#!\"\"" }{TEXT 523 2 ")) " }{TEXT -1 5 " are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "`de t(A^3)` := det(evalm(A^3)) : `det(A^(-2))` := det(evalm(A^(-2))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Det(A^3) = `det(A^3)` ; \+ Det(A^(`-2`)) = `det(A^(-2))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %$DetG6#*$)%\"AG\"\"$\"\"\"!&_>#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %$DetG6#)%\"AG%#-2G#\"\"\"\"$%y" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "(b) The values of " }{TEXT 404 4 "(Det" }{TEXT -1 1 "[" }{TEXT 403 1 "A" }{TEXT -1 1 "]" }{TEXT 473 2 ")^" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT -1 7 " and " }{TEXT 525 4 "(Det" }{TEXT -1 1 "[" }{TEXT 524 1 "A" }{TEXT -1 1 "]" }{TEXT 526 3 ")^(" }{XPPEDIT 18 0 "-2" "6#,$\"\"#!\"\"" }{TEXT 527 1 ")" } {TEXT -1 5 " are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`(det( A))^3` := (det(A))^3 : `(det(A))^(-2)` := (det(A))^(-2) :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "[Det(A)]^3 = `(det(A))^3` ; [Det(A)]^(`-2`) = `(det(A))^(-2)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)7#-%$DetG6#%\"AG\"\"$\"\"\"!&_>#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)7#-%$DetG6#%\"AG%#-2G#\"\"\"\"$%y" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "[ For " }{TEXT 553 22 "integer exponentiation" }{TEXT -1 36 " of matrices, refer to \+ Unit (15). ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 271 "" 0 "" {TEXT 393 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 4 "N.B." }{TEXT -1 133 " The de terminant of the product of any square matrices of the same order is e qual to the product of the determinants of the matrices" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 376 4 "Det(" }{TEXT -1 1 "[" }{TEXT 271 1 "A" }{TEXT -1 3 "] [" }{TEXT 272 1 "B" }{TEXT -1 1 "]" }{TEXT 474 7 ") = D et" }{TEXT -1 1 "[" }{TEXT 273 1 "A" }{TEXT -1 2 "] " }{TEXT 377 3 "De t" }{TEXT -1 1 "[" }{TEXT 274 1 "B" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Let " } {TEXT 475 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 476 3 " \327 " } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 477 1 ")" }{TEXT -1 12 " matrices \+ [" }{TEXT 275 1 "A" }{TEXT -1 7 "] and [" }{TEXT 276 1 "B" }{TEXT -1 13 "] be given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "A := \+ matrix(2, 2, [5, 4, -8, 7]) : B := matrix(2, 2, [3, 2, 5, 4]) : A \+ = matrix(A) ; B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% \"AG-%'matrixG6#7$7$\"\"&\"\"%7$!\")\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7$7$\"\"$\"\"#7$\"\"&\"\"%" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "(a) The determinant of the matrix product, " }{TEXT 378 4 "Det (" }{TEXT -1 1 "[" }{TEXT 286 1 "A" }{TEXT -1 3 "] [" }{TEXT 287 1 "B " }{TEXT -1 1 "]" }{TEXT 478 1 ")" }{TEXT -1 16 ", has the value" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "`det(AB)` := det(multiply(A, B)) : Det(A * B) = `det(AB)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%$DetG6#*&%\"AG\"\"\"%\"BGF)\"$M\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "(b) The product " } {TEXT 379 3 "Det" }{TEXT -1 1 "[" }{TEXT 277 1 "A" }{TEXT -1 2 "] " } {TEXT 380 3 "Det" }{TEXT -1 1 "[" }{TEXT 278 1 "B" }{TEXT -1 27 "] is the following number:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "` det(A) det(B)` := det(A) * det(B) : Det(A) * Det(B) = `det(A) det(B) ` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$DetG6#%\"AG\"\"\"-F&6#%\" BGF)\"$M\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 279 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 292 4 "N.B." }{TEXT -1 26 " The det erminant of the " }{TEXT 293 9 "transpose" }{TEXT -1 8 " of a " } {TEXT 300 13 "square matrix" }{TEXT -1 43 " is equal to the determina nt of the matrix" }}}{EXCHG {PARA 263 "" 0 "" {TEXT 361 10 "Det(Transp " }{TEXT -1 1 "[" }{TEXT 298 1 "A" }{TEXT -1 1 "]" }{TEXT 479 7 ") = D et" }{TEXT -1 1 "[" }{TEXT 299 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exempla rily, consider a " }{TEXT 480 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 302 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 481 1 ")" } {TEXT -1 10 " matrix [" }{TEXT 301 1 "A" }{TEXT -1 10 "] given as" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "A := matrix(3, 3, [1, -2, 3 , 0, 4, -2, 6, -1, -1]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"\"!\"#\"\"$7%\"\"!\"\"%F+7%\" \"'!\"\"F2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "(a) The determinant of the matrix, " }{TEXT 362 3 "Det" }{TEXT -1 1 "[" }{TEXT 303 1 "A" }{TEXT -1 17 "], has the val ue" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : Det(A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\" AG!#a" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "(b) The determinant of the matrix transpose, " } {TEXT 363 10 "Det(Transp" }{TEXT -1 1 "[" }{TEXT 304 1 "A" }{TEXT -1 1 "]" }{TEXT 482 1 ")" }{TEXT -1 16 ", has the value" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "`det(transp(A))` := det(transpose(A )) : Det(Transp(A)) = `det(transp(A))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#-%'TranspG6#%\"AG!#a" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 264 "" 0 "" {TEXT 294 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 589 4 "N.B." }{TEXT -1 86 " Transposition of the following matrix doe s not change the determinant of the matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "V := matrix(3, 3, [1, 1, 1, a, b, c, a^2, b^2, c^2 ]) : V = matrix(V) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"VG-%'mat rixG6#7%7%\"\"\"F*F*7%%\"aG%\"bG%\"cG7%*$)F,\"\"#F**$)F-F2F**$)F.F2F* " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "(a) The determinant of the matrix, " }{TEXT 592 3 "Det" }{TEXT -1 1 "[" }{TEXT 591 1 "V" }{TEXT -1 17 "], has the value" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`det(V)` := factor(det(V)) \+ : Det(V) = `det(V)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#% \"VG,$*(,&%\"cG!\"\"%\"bG\"\"\"F.,&%\"aGF.F+F,F.,&F0F.F-F,F.F," }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "(b) The determinant of the matrix transpose, " }{TEXT 594 10 " Det(Transp" }{TEXT -1 1 "[" }{TEXT 593 1 "V" }{TEXT -1 1 "]" }{TEXT 595 1 ")" }{TEXT -1 16 ", has the value" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "`det(transp(V))` := factor(det(transpose(V))) : Det (Transp(V)) = `det(transp(V))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %$DetG6#-%'TranspG6#%\"VG,$*(,&%\"cG!\"\"%\"bG\"\"\"F1,&%\"aGF1F.F/F1, &F3F1F0F/F1F/" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "The determinant of this matrix is called \+ an " }{TEXT 596 9 "alternate" }{TEXT -1 18 " determinant or " } {TEXT 597 9 "alternant" }{TEXT -1 46 ". The matrix itself is the tran spose of the " }{TEXT 598 11 "Vandermonde" }{TEXT -1 52 " matrix and may be created automatically using the " }{TEXT 599 11 "vandermonde" }{TEXT -1 15 " function, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "V := transpose(vandermonde([a, b, c])) : V = matrix(V) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"VG-%'matrixG6#7%7%\"\"\"F*F*7%%\"a G%\"bG%\"cG7%*$)F,\"\"#F**$)F-F2F**$)F.F2F*" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 294 "" 0 "" {TEXT 590 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 331 4 "N.B." }{TEXT -1 84 " If the elements of a row (or of a column) of a matrix are multiplied by a scalar " }{XPPEDIT 18 0 "lambda" "6# %'lambdaG" }{TEXT -1 69 ", then the value of the determinant of the m atrix is multiplied by " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" } {TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Exemplarily, consider the same " }{TEXT 483 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 349 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 484 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 332 1 "A" }{TEXT -1 37 "] as above and assume that a scalar \+ " }{XPPEDIT 18 0 "lambda=3" "6#/%'lambdaG\"\"$" }{TEXT -1 1 "." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "lambda := 3 :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "(a) Let the column " } {XPPEDIT 18 0 "j=2" "6#/%\"jG\"\"#" }{TEXT -1 20 " be multiplied by \+ " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 30 ". This yields t he following " }{TEXT 485 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 333 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 486 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 334 1 "B" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "j := 2 : B := mulcol(A, j, lambda) : \+ B = matrix(B) ;" }}{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 32 "The determ inant of the matrix, " }{TEXT 364 3 "Det" }{TEXT -1 1 "[" }{TEXT 335 1 "B" }{TEXT -1 17 "], has the value" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(B)` := det(B) : Det(B) = `det(B)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"BG!$i\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "which is " }{TEXT 355 5 "three" }{TEXT -1 47 " times that of the determinant of \+ the matrix [" }{TEXT 336 1 "A" }{TEXT -1 9 "], i.e. " }{TEXT 365 3 "D et" }{TEXT -1 1 "[" }{TEXT 337 1 "B" }{TEXT -1 1 "]" }{TEXT 487 3 " = \+ " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT -1 1 " " }{TEXT 366 3 "Det" } {TEXT -1 1 "[" }{TEXT 338 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "(b) Let the \+ row " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 20 " be multip lied by " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 30 ". This yields the following " }{TEXT 488 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$ " }{TEXT 339 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 489 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 340 1 "C" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "i := 1 : C := mulrow(A, i, lambda ) : C = matrix(C) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"CG-%'matr ixG6#7%7%\"\"$!\"'\"\"*7%\"\"!\"\"%!\"#7%\"\"'!\"\"F3" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "T he determinant of the matrix, " }{TEXT 367 3 "Det" }{TEXT -1 1 "[" } {TEXT 341 1 "C" }{TEXT -1 17 "], has the value" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 42 "`det(C)` := det(C) : Det(C) = `det(C)` ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"CG!$i\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "which i s " }{TEXT 356 5 "three" }{TEXT -1 47 " times that of the determinan t of the matrix [" }{TEXT 342 1 "A" }{TEXT -1 9 "], i.e. " }{TEXT 368 3 "Det" }{TEXT -1 1 "[" }{TEXT 344 1 "C" }{TEXT -1 1 "]" }{TEXT 490 3 " = " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT -1 1 " " }{TEXT 369 3 "Det" }{TEXT -1 1 "[" }{TEXT 343 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 269 "" 0 "" {TEXT 345 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 295 4 "N.B." }{TEXT -1 24 " The determinant of a " } {TEXT 305 29 "matrix multiplied by a scalar" }{TEXT -1 143 " (whether real or complex) is equal to the product of the matrix determinant an d the scalar raised to the power equal to the number of rows, " } {XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT -1 16 ", or columns, " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 16 ", in the matrix" }}}{EXCHG {PARA 265 "" 0 "" {TEXT 370 3 "Det" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "k" "6#% \"kG" }{TEXT -1 2 " [" }{TEXT 297 1 "A" }{TEXT -1 3 "]) " }{TEXT 347 1 "=" }{TEXT -1 1 " " }{XPPEDIT 18 0 "k^m" "6#)%\"kG%\"mG" }{TEXT -1 1 " " }{TEXT 371 3 "Det" }{TEXT -1 1 "[" }{TEXT 296 1 "A" }{TEXT -1 2 "] " }{TEXT 357 1 "=" }{TEXT -1 1 " " }{XPPEDIT 18 0 "k^n" "6#)%\"kG% \"nG" }{TEXT -1 1 " " }{TEXT 372 3 "Det" }{TEXT -1 1 "[" }{TEXT 358 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Exemplarily, consider the same " }{TEXT 491 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 348 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 492 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 306 1 "A" }{TEXT -1 38 "] as before and assume that a scalar \+ " }{XPPEDIT 18 0 "k=2" "6#/%\"kG\"\"#" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "k := 2 :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "(a) The determinant of the matrix multipl ied by the scalar, " }{TEXT 373 4 "Det(" }{XPPEDIT 18 0 "k" "6#%\"kG " }{TEXT -1 2 " [" }{TEXT 346 1 "A" }{TEXT -1 1 "]" }{TEXT 493 1 ")" } {TEXT -1 16 ", has the value" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "`det(kA)` := det(k*A) : k := 'k' : Det(k * A) = `det(kA)` ; k := 2 :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#*&%\"kG\"\"\"% \"AGF)!$K%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 312 39 "Notice that the multiplication order, " }{TEXT 309 3 "k A" }{TEXT 313 6 " or " }{TEXT 310 3 "A k" }{TEXT -1 2 ", " }{TEXT 314 12 " under the " }{TEXT 311 3 "det" }{TEXT -1 1 " " } {TEXT 315 13 " function is " }{TEXT 308 3 "not" }{TEXT 316 10 " import ant" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "(b) The product of the matrix det erminant and the scalar raised to the power equal to the number of row s, " }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT -1 16 ", or columns, " } {XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 19 ", in the matrix, " } {XPPEDIT 18 0 "k^m" "6#)%\"kG%\"mG" }{TEXT -1 1 " " }{TEXT 374 3 "Det " }{TEXT -1 1 "[" }{TEXT 359 1 "A" }{TEXT -1 8 "] and " }{XPPEDIT 18 0 "k^n" "6#)%\"kG%\"nG" }{TEXT -1 1 " " }{TEXT 375 3 "Det" }{TEXT -1 1 "[" }{TEXT 307 1 "A" }{TEXT -1 84 "], respectively, may be compu ted using either of the following alternative methods:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "`k^m det(A)` := k^rowdim(A)*det(A) \+ : k := 'k' : k^m * Det(A) = `k^m det(A)` ; k := 2 :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%\"kG%\"mG\"\"\"-%$DetG6#%\"AGF(!$K%" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "`k^n det(A)` := k^coldim(A)*det(A) : k := 'k' : k ^n * Det(A) = `k^n det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)% \"kG%\"nG\"\"\"-%$DetG6#%\"AGF(!$K%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The value of either pro duct, " }{XPPEDIT 18 0 "k^m" "6#)%\"kG%\"mG" }{TEXT -1 1 " " }{TEXT 381 3 "Det" }{TEXT -1 1 "[" }{TEXT 360 1 "A" }{TEXT -1 7 "] or " } {XPPEDIT 18 0 "k^n" "6#)%\"kG%\"nG" }{TEXT -1 1 " " }{TEXT 382 3 "Det " }{TEXT -1 1 "[" }{TEXT 352 1 "A" }{TEXT -1 8 "], is " }{TEXT 354 5 "eight" }{TEXT -1 9 " times " }{TEXT 494 2 "( " }{XPPEDIT 18 0 "k^ 3=8" "6#/*$%\"kG\"\"$\"\")" }{TEXT 495 2 " )" }{TEXT -1 41 " that of \+ the determinant of the matrix [" }{TEXT 350 1 "A" }{TEXT -1 9 "], i.e. " }{TEXT 383 4 "Det(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT -1 2 " [" }{TEXT 353 1 "A" }{TEXT -1 1 "]" }{TEXT 496 4 ") = " }{XPPEDIT 18 0 "8 " "6#\"\")" }{TEXT -1 1 " " }{TEXT 384 3 "Det" }{TEXT -1 1 "[" }{TEXT 351 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 319 15 "The functions " }{TEXT 317 6 "co ldim" }{TEXT 320 7 " and " }{TEXT 318 6 "rowdim" }{TEXT 321 64 " re turn the number of columns or rows in a matrix, respectively" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 266 " " 0 "" {TEXT 322 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 411 4 "N.B." }{TEXT -1 15 " If a matrix \+ [" }{TEXT 408 1 "B" }{TEXT -1 27 "] is formed from a matrix [" }{TEXT 409 1 "A" }{TEXT -1 47 "] by interchanging two rows or two columns of \+ [" }{TEXT 410 1 "A" }{TEXT -1 7 "], then" }}}{EXCHG {PARA 274 "" 0 "" {TEXT 422 3 "Det" }{TEXT -1 1 "[" }{TEXT 420 1 "B" }{TEXT -1 1 "]" } {TEXT 497 7 " = \226Det" }{TEXT -1 1 "[" }{TEXT 421 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 498 1 "(" }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 499 3 " \327 " }{XPPEDIT 18 0 "4" "6#\"\"%" } {TEXT 500 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 412 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "A := matr ix(4, 4, [-2, 4, 3, -3, -2, -3, 6, -2, -5, 3, -7, -9, -2, -3, 4, -8]) \+ : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrix G6#7&7&!\"#\"\"%\"\"$!\"$7&F*F-\"\"'F*7&!\"&F,!\"(!\"*7&F*F-F+!\")" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "(a) The determinant of [" }{TEXT 413 1 "A" }{TEXT -1 4 "] is" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : Det (A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG\"% +9" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "(b) The matrix [" }{TEXT 414 1 "B" }{TEXT -1 35 "] formed by interchanging columns " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT -1 6 " of [" }{TEXT 416 1 "A" }{TEXT -1 27 "], and the determinant of [" }{TEXT 417 1 "B" }{TEXT -1 5 "] are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "B := \+ swapcol(A, 2, 4) : B = matrix(B) ; `det(B)` := det(B) : Det(B) = `det(B)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7&7&! \"#!\"$\"\"$\"\"%7&F*F*\"\"'F+7&!\"&!\"*!\"(F,7&F*!\")F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"BG!%+9" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "(c) The matrix [" }{TEXT 415 1 "B" }{TEXT -1 32 "] formed by interchanging rows " } {XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT -1 6 " of [" }{TEXT 418 1 "A" }{TEXT -1 27 "], and t he determinant of [" }{TEXT 419 1 "B" }{TEXT -1 5 "] are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "B := swaprow(A, 1, 3) : B = matri x(B) ; `det(B)` := det(B) : Det(B) = `det(B)` ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%\"BG-%'matrixG6#7&7&!\"&\"\"$!\"(!\"*7&!\"#!\"$\"\" 'F/7&F/\"\"%F+F07&F/F0F3!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$De tG6#%\"BG!%+9" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "This property is sometimes expressed in t he following words:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "\"If two rows (or columns) of a ma trix are interchanged, the determinant changes sign.\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 273 "" 0 "" {TEXT 423 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 433 4 "N.B." }{TEXT -1 15 " If a matrix [" }{TEXT 434 1 "B " }{TEXT -1 34 "] is formed from a square matrix [" }{TEXT 435 1 "A" } {TEXT -1 39 "] by adding to one row (or column) of [" }{TEXT 436 1 "A " }{TEXT -1 45 "] a scalar times another row (or column) of [" }{TEXT 437 1 "A" }{TEXT -1 7 "], then" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 431 3 "Det" }{TEXT -1 1 "[" }{TEXT 429 1 "B" }{TEXT -1 1 "]" }{TEXT 501 6 " = Det" }{TEXT -1 1 "[" }{TEXT 430 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "E xemplarily, consider a " }{TEXT 502 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$ " }{TEXT 503 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 504 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 432 1 "A" }{TEXT -1 10 "] given as" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "A := matrix(3, 3, [4, -2, \+ 3, 0, 4, -2, 6, -1, -3]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"%!\"#\"\"$7%\"\"!F*F+7%\"\"'! \"\"!\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 24 "(a) The determinant of [" }{TEXT 438 1 "A" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := d et(A) : Det(A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$D etG6#%\"AG!$/\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "(b) Let each element of the column " } {XPPEDIT 18 0 "j[B]=2" "6#/&%\"jG6#%\"BG\"\"#" }{TEXT -1 19 " in a ne w matrix [" }{TEXT 424 1 "B" }{TEXT -1 73 "] be formed as a sum of the corresponding element in the same column of [" }{TEXT 425 1 "A" } {TEXT -1 14 "] and scalar " }{XPPEDIT 18 0 "lambda=k^2" "6#/%'lambdaG *$%\"kG\"\"#" }{TEXT -1 49 " times the corresponding element of the c olumn " }{XPPEDIT 18 0 "j[A]=1" "6#/&%\"jG6#%\"AG\"\"\"" }{TEXT -1 6 " of [" }{TEXT 426 1 "A" }{TEXT -1 9 "], i.e. " }{XPPEDIT 18 0 "b[i, j[B]] = a[i,j[A]] + lambda*a[i, j[A]]" "6#/&%\"bG6$%\"iG&%\"jG6#%\"BG ,&&%\"aG6$F'&F)6#%\"AG\"\"\"*&%'lambdaGF3&F.6$F'&F)6#F2F3F3" }{TEXT 427 1 "." }{TEXT -1 7 " Let " }{XPPEDIT 18 0 "k= 4" "6#/%\"kG\"\"%" }{TEXT 428 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "j[A] := 1 : j[B] := 2 : k := 4 \+ : lambda := k^2 :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Therefore, the matrix [" }{TEXT 439 1 "B" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "B := addcol(A, j[A], j[B], lambda) : B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7%7%\"\"%\"#i\"\"$7%\"\"!F *!\"#7%\"\"'\"#&*!\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "(c) The determinant of [" }{TEXT 440 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 0 "" 0 "" {TEXT -1 33 "(d) Let each element of the row " }{XPPEDIT 18 0 "i[B]=1" "6#/&%\"iG6#%\"BG\"\"\"" }{TEXT -1 19 " in a new matrix [" }{TEXT 441 1 "B" }{TEXT -1 70 "] be formed as a sum of the corresponding element in the same row of [" }{TEXT 442 1 "A" }{TEXT -1 14 "] and scalar " }{XPPEDIT 18 0 "lambda=sin(Pi/ k)" "6#/%'lambdaG-%$sinG6#*&%#PiG\"\"\"%\"kG!\"\"" }{TEXT -1 46 " tim es the corresponding element of the row " }{XPPEDIT 18 0 "i[A]=3" "6# /&%\"iG6#%\"AG\"\"$" }{TEXT -1 6 " of [" }{TEXT 443 1 "A" }{TEXT -1 9 "], i.e. " }{XPPEDIT 18 0 "b[i[B],j]=a[i[A],j]+lambda*a[i[A],j]" "6 #/&%\"bG6$&%\"iG6#%\"BG%\"jG,&&%\"aG6$&F(6#%\"AGF+\"\"\"*&%'lambdaGF3& F.6$&F(6#F2F+F3F3" }{TEXT 445 1 "." }{TEXT -1 7 " Let " }{XPPEDIT 18 0 "k=6" "6#/%\"kG\"\"'" }{TEXT 444 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "i[A] := 3 \+ : i[B] := 1 : k := 6 : lambda := sin(Pi/k) :" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Therefore, the matrix [" }{TEXT 446 1 "B" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 54 "B := addrow(A, i[A], i[B], lambda) : B = m atrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7%7%\" \"(#!\"&\"\"##\"\"$F-7%\"\"!\"\"%!\"#7%\"\"'!\"\"!\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "( e) The determinant of [" }{TEXT 447 1 "B" }{TEXT -1 4 "] is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(B)` := det(B) : Det(B ) = `det(B)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"BG!$/\" " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "The determinants of (a), (c), and (e) are identical." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 200 "An obvious implication of the above result is that if a row (o r column) of a matrix is expressible as the sum of multiples of other \+ rows (or columns) of the matrix, then the value of the determinant " }{TEXT 19 4 "must" }{TEXT -1 6 " be " }{XPPMATH 20 "6#%%zeroG" } {TEXT -1 163 ". This is so because by subtraction of this sum of mult iples of other rows (or columns) in question, it is possible to produc e a row (or column) containing only " }{XPPMATH 20 "6#%%zeroG" } {TEXT -1 11 " elements." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a " }{TEXT 540 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 541 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 542 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 539 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "A := matrix(3, 3, [6, 18, 8, 1, 5, 7, 3, 9, 4]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6 #7%7%\"\"'\"#=\"\")7%\"\"\"\"\"&\"\"(7%\"\"$\"\"*\"\"%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "N otice that the elements of the first row of [" }{TEXT 545 1 "A" } {TEXT -1 53 "] are equal to the doubled elements of the third row." }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Consequently, the determinant of [" }{TEXT 543 1 "A" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det( A) : Det(A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG 6#%\"AG\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 286 "" 0 "" {TEXT 529 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 601 4 "N.B." }{TEXT -1 41 " The det erminant of the following matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "A := matrix(3, 3, [sin(t+Pi/4), sin(t), cos(t), sin( t+Pi/4), cos(t), sin(t), 1, a, 1-a]) : A = matrix(A) ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%-%$sinG6#,&%\"tG\"\"\"*&#F /\"\"%F/%#PiGF/F/-F+6#F.-%$cosGF57%F*F6F47%F/%\"aG,&F/F/F:!\"\"" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "is independent of " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 39 " and is expressible as a function of " }{XPPEDIT 18 0 "t" "6#%\"tG " }{TEXT -1 12 " only, i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : Det(A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG,**&-%$sinG6#,&%\"tG\"\"\"*&#F/\"\"%F/%# PiGF/F/F/-%$cosG6#F.F/F/*&F*F/-F+F6F/!\"\"*$)F8\"\"#F/F/*$)F4F " 0 "" {MPLTEXT 1 0 78 "`det(A)` : = collect(combine(expand(det(A))), cos(2*t)) : Det(A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG*&,&*$-%%sqrtG6#\"\"# \"\"\"#F/F.F/!\"\"F/-%$cosG6#,$%\"tGF.F/" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 295 "" 0 "" {TEXT 600 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 572 4 "N.B." }{TEXT -1 107 " There are symmetric matrices with elemen ts created according to a specific pattern whose determinant is " } {XPPMATH 20 "6#%%zeroG" }{TEXT -1 39 ". One of them is the following \+ matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "A := matrix(3, 3 , [a^2, a*b, a*c, a*b, b^2, b*c, a*c, b*c, c^2]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%*$)%\"aG\"\"# \"\"\"*&F,F.%\"bGF.*&F,F.%\"cGF.7%F/*$)F0F-F.*&F0F.F2F.7%F1F6*$)F2F-F. " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "From the definition of the determinant, it can be easily \+ noticed that the determinant of [" }{TEXT 574 1 "A" }{TEXT -1 6 "] is \+ " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 50 ", which is verified by the \+ following computation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "` det(A)` := det(A) : Det(A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 143 "This may not be notice able so easily if such a symmetric matrix is given with numerical elem ents following the same pattern. For example, let " }{XPPEDIT 18 0 "a " "6#%\"aG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 8 ", and " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 23 " in the above matrix [" }{TEXT 575 1 "A" }{TEXT -1 13 "] be given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "a := 3*exp(2) : b := -5 : c := \+ 7*sin(Pi/3) : 'a' = a ; 'b' = b ; 'c' = c ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"aG,$-%$expG6#\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"bG!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG,$* $-%%sqrtG6#\"\"$\"\"\"#\"\"(\"\"#" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Evaluation of [" }{TEXT 576 1 "A" }{TEXT -1 24 "] with these values of " }{XPPEDIT 18 0 "a" " 6#%\"aG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 8 ", and " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 46 " yields the follo wing exact symmetric matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "A := map(combine, map(x->eval(x), A)) : A = matrix(A) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%,$-%$expG6#\"\"% \"\"*,$-F,6#\"\"#!#:,$*&F1\"\"\"-%%sqrtG6#\"\"$F7#\"#@F37%F0\"#D,$*$F8 F7#!#NF37%F5F@#\"$Z\"F." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The determinant of the exact [" } {TEXT 577 1 "A" }{TEXT -1 16 "] is precisely " }{XPPMATH 20 "6#%%zero G" }{TEXT -1 7 ", viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " `det(A)` := det(A) : Det(A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "while the determinant o f [" }{TEXT 578 1 "A" }{TEXT -1 60 "] subjected to floating-point eval uation yielding the matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "A := evalf(matrix(A)) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%$\"+.N$Q\"\\!\"($!+:%e$36F,$\"+iJ \"QM\"F,7%F-$\"#D\"\"!$!+9*)3JI!\")7%F/F5$\"++++vOF7" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "is only approximately equal to " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 7 ", vi z." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : Det(A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\" AG$!*pYlv%!#8" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 292 "" 0 "" {TEXT 573 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 528 4 "N.B." }{TEXT -1 50 " T he determinant of a diagonal matrix having no " }{XPPMATH 20 "6#%%zer oG" }{TEXT -1 70 " elements on the diagonal equals the product of its diagonal elements" }}}{EXCHG {PARA 287 "" 0 "" {TEXT 537 3 "Det" } {TEXT -1 1 "[" }{TEXT 536 1 "A" }{TEXT -1 1 "]" }{TEXT 538 3 " = " } {XPPEDIT 18 0 "Product(diag_el[i](A),i=1..n)" "6#-%(ProductG6$-&%(diag _elG6#%\"iG6#%\"AG/F*;\"\"\"%\"nG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Exemplarily, consider a \+ " }{TEXT 532 1 "(" }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 531 3 " \327 " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 533 1 ")" }{TEXT -1 19 " diagonal matrix [" }{TEXT 530 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "A := diag(3, -4, 2, -5) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7&7&\"\"$\"\"! F+F+7&F+!\"%F+F+7&F+F+\"\"#F+7&F+F+F+!\"&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "(a) The determina nt of [" }{TEXT 534 1 "A" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : Det(A) = `det(A)` ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG\"$?\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "( b) The product of the diagonal elements of [" }{TEXT 535 1 "A" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "`product(diag _el(A))` := 1 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "for i t o rowdim(A) do for j to coldim(A) do if j = i then `product(diag_ el(A))` := `product(diag_el(A))` * A[i,j] fi : od : od :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "`product(diag_el(A))` := `pr oduct(diag_el(A))` : i := 'i' :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Product(diag_el[i](A), i=1..rowdim(A)) = `product(dia g_el(A))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(ProductG6$-&%(diag_ elG6#%\"iG6#%\"AG/F+;\"\"\"\"\"%\"$?\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 275 "" 0 "" {TEXT 448 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 587 4 "N.B." }{TEXT -1 82 " The determinant of a matrix equals the pr oduct of the eigenvalues of the matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "[ For the " } {TEXT 588 11 "eigenvalues" }{TEXT -1 36 " of a matrix, refer to Unit \+ (21). ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 293 " " 0 "" {TEXT 586 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 387 4 "N.B." }{TEXT -1 43 " If the deter minant of a matrix of order " }{TEXT 505 1 "(" }{XPPEDIT 18 0 "3" "6# \"\"$" }{TEXT 506 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 507 1 ")" }{TEXT -1 87 " or higher is to be evaluated manually, it is nec essary to introduce the notion of a " }{TEXT 385 5 "minor" }{TEXT -1 7 " and " }{TEXT 407 8 "cofactor" }{TEXT -1 4 " ( " }{TEXT 386 12 " signed minor" }{TEXT -1 32 " ) of an element of the matrix." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "[ For the definitions of the " }{TEXT 554 5 "minor" }{TEXT -1 7 " and " }{TEXT 555 8 "cofactor" }{TEXT -1 24 ", refer to Unit (12 ). ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 388 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 265 4 "N.B." }{TEXT -1 38 " In computing determinants of large " }{TEXT 266 6 "sparse" }{TEXT -1 39 " matric es, it is advisable to use the " }{TEXT 285 6 "sparse" }{TEXT -1 65 " \+ optional directive. This argument specifies that the method of " } {TEXT 267 15 "minor expansion" }{TEXT -1 32 " should be used by the p rogram." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "[ Refer also to Unit (12) for the " }{TEXT 552 7 "L aplace" }{TEXT -1 22 " expansion theorem. ]" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Exemplarily, co mpute the determinant of a " }{TEXT 509 1 "(" }{XPPEDIT 18 0 "6" "6# \"\"'" }{TEXT 508 3 " \327 " }{XPPEDIT 18 0 "6" "6#\"\"'" }{TEXT 510 1 ")" }{TEXT -1 17 " sparse matrix [" }{TEXT 268 1 "E" }{TEXT -1 10 " ] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "E := array(1 ..6, 1..6, [(1,1)=15, (1,2)=-10, (1,3)=-5, (2,1)=-10, (2,2)=10, (3,1)= -5, (3,3)=0, (3,4)=-12, (3,5)=0, (4,3)=-12, (4,4)=12, (5,3)=-8, (5,5)= 23, (5,6)=0, (6,5)=-15, (6,6)=15], sparse) : E = matrix(E) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"EG-%'matrixG6#7(7(\"#:!#5!\"&\"\"! F-F-7(F+\"#5F-F-F-F-7(F,F-F-!#7F-F-7(F-F-F1\"#7F-F-7(F-F-!\")F-\"#BF-7 (F-F-F-F-!#:F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The determinant of [" }{TEXT 291 1 "E" } {TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`det(E) ` := det(E, sparse) : Det(E) = `det(E)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"EG!(+!>N" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 288 "" 0 "" {TEXT 544 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "P roceed to Unit (12) for \"" }{TEXT 517 42 "The minor and cofactor of a matrix element" }{TEXT -1 2 "\"." }}}{EXCHG {PARA 262 "" 0 "" {TEXT 516 67 "-------------------------------------------------------------- -----" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }