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Blank" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 " Matrices" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 50 "A matrix is a rectangular array. We refer to the \+ " }{TEXT 256 4 "size" }{TEXT -1 18 " of a matrix as " }{TEXT 258 2 " r " }{TEXT 257 1 "x" }{TEXT 259 2 " c" }{TEXT -1 14 " if it has " } {TEXT 260 1 "r" }{TEXT -1 12 " rows and " }{TEXT 261 1 "c" }{TEXT -1 19 " columns. \n\nIf " }{TEXT 262 7 "r = c" }{TEXT -1 35 " \+ then we say that the matrix is " }{TEXT 263 6 "square" }{TEXT -1 5 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{TEXT 264 1 "A" }{TEXT -1 47 " is a matrix then we refer to the \+ entry in the" }{TEXT 266 2 " " }{XPPEDIT 267 1 "i^th" ")%\"iG%#thG" } {TEXT -1 8 " row, " }{TEXT 268 1 " " }{XPPEDIT 269 1 "j^th" ")%\"jG% #thG" }{TEXT -1 13 " column as " }{TEXT 265 10 "A[i , j] " }{TEXT -1 7 " or " }{TEXT 270 1 " " }{XPPEDIT 271 1 "A[i,j]" "&%\"AG6$%\"i G%\"jG" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 13 "The elements " }{TEXT 275 1 " " }{XPPEDIT 276 1 "A [1,1]" "&%\"AG6$\"\"\"F%" }{TEXT -1 3 " , " }{TEXT 277 1 " " } {XPPEDIT 278 1 "A[2,2]" "&%\"AG6$\"\"#F%" }{TEXT -1 10 " , ... , " } {TEXT 279 1 " " }{XPPEDIT 280 1 "A[m,m]" "&%\"AG6$%\"mGF%" }{TEXT -1 2 " " }{TEXT 281 16 "( m = min(r,c) )" }{TEXT -1 29 " are said to con stitute the " }{TEXT 272 13 "main diagonal" }{TEXT -1 5 " or " } {TEXT 273 18 "principal diagonal" }{TEXT -1 5 " of " }{TEXT 274 1 "A " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{TEXT 285 1 "A" }{TEXT -1 9 " is a " }{TEXT 283 2 "r " }{TEXT 282 1 "x" }{TEXT 284 2 " c" }{TEXT -1 20 " matrix the n the " }{TEXT 287 3 " c " }{TEXT 286 1 "x" }{TEXT 288 2 " r" }{TEXT -1 14 " matrix whose" }{TEXT 289 2 " " }{XPPEDIT 290 1 "i^th" ")%\"i G%#thG" }{TEXT -1 11 " row is the" }{TEXT 291 2 " " }{XPPEDIT 292 1 " i^th" ")%\"iG%#thG" }{TEXT -1 12 " column of " }{TEXT 293 1 "A" } {TEXT -1 18 " is called the " }{TEXT 295 9 "transpose" }{TEXT -1 4 " of " }{TEXT 294 1 "A" }{TEXT -1 18 " . It is denoted " }{TEXT 296 1 " " }{XPPEDIT 297 1 "A^t" ")%\"AG%\"tG" }{TEXT -1 16 ". Notice tha t " }{TEXT 300 2 " " }{XPPEDIT 301 1 "(A^t)*[i,j]=A[j,i]" "/*&)%\"AG% \"tG\"\"\"7$%\"iG%\"jGF'&F%6$F*F)" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "To use matrices in " }{TEXT 298 5 "Maple" }{TEXT -1 49 " first load the linear algebra package ca lled " }{TEXT 299 6 "linalg" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg);" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warn ing, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, \+ new definition for trace" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7^r%.Block DiagonalG%,GramSchmidtG%,JordanBlockG%)LUdecompG%)QRdecompG%*Wronskian G%'addcolG%'addrowG%$adjG%(adjointG%&angleG%(augmentG%(backsubG%%bandG %&basisG%'bezoutG%,blockmatrixG%(charmatG%)charpolyG%)choleskyG%$colG% 'coldimG%)colspaceG%(colspanG%*companionG%'concatG%%condG%)copyintoG%* crossprodG%%curlG%)definiteG%(delcolsG%(delrowsG%$detG%%diagG%(diverge G%(dotprodG%*eigenvalsG%,eigenvaluesG%-eigenvectorsG%+eigenvectsG%,ent ermatrixG%&equalG%,exponentialG%'extendG%,ffgausselimG%*fibonacciG%+fo rwardsubG%*frobeniusG%*gausselimG%*gaussjordG%(geneqnsG%*genmatrixG%%g radG%)hadamardG%(hermiteG%(hessianG%(hilbertG%+htransposeG%)ihermiteG% *indexfuncG%*innerprodG%)intbasisG%(inverseG%'ismithG%*issimilarG%'isz eroG%)jacobianG%'jordanG%'kernelG%*laplacianG%*leastsqrsG%)linsolveG%' mataddG%'matrixG%&minorG%(minpolyG%'mulcolG%'mulrowG%)multiplyG%%normG %*normalizeG%*nullspaceG%'orthogG%*permanentG%&pivotG%*potentialG%+ran dmatrixG%+randvectorG%%rankG%(ratformG%$rowG%'rowdimG%)rowspaceG%(rows panG%%rrefG%*scalarmulG%-singularvalsG%&smithG%&stackG%*submatrixG%*su bvectorG%)sumbasisG%(swapcolG%(swaprowG%*sylvesterG%)toeplitzG%&traceG %*transposeG%,vandermondeG%*vecpotentG%(vectdimG%'vectorG%*wronskianG " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 143 "A matrix is entered as a list of rows. E ach row is an element of the list of rows. Each row is itself a list. \+ The next example will illustrate:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "A := matrix( [ [1,2,3,4], [5,6,7,8], [9,10,11,12] ] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7&\"\"\"\"\"#\"\"$\"\"%7 &\"\"&\"\"'\"\"(\"\")7&\"\"*\"#5\"#6\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "A[2,3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "transpose(A);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7%\"\"\"\"\"&\"\"*7%\"\"#\"\"'\"#5 7%\"\"$\"\"(\"#67%\"\"%\"\")\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 " Determinants - The definition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 145 "We will assume now that the entries of our matrices can be multiplied together and \+ added. In this case we can define an important function, the " }{TEXT 302 11 "determinant" }{TEXT -1 22 ", on the set of all " }{TEXT 303 6 "square" }{TEXT -1 41 " matrices. The definition is inductive." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " } {TEXT 304 1 " " }{XPPEDIT 305 1 "A = MATRIX([[a]])" "/%\"AG-%'MATRIXG6 #7#7#%\"aG" }{TEXT -1 33 " then we define the determinant " }{TEXT 306 1 " " }{XPPEDIT 307 1 "det(A) " "-%$detG6#%\"AG" }{TEXT -1 5 " of \+ " }{TEXT 308 1 " " }{XPPEDIT 309 1 "A" "I\"AG6\"" }{TEXT -1 8 " by \+ " }{TEXT 310 1 " " }{XPPEDIT 311 1 "det(A) =a" "/-%$detG6#%\"AG%\"aG " }{TEXT -1 34 " . This defines determinants of " }{TEXT 313 3 " 1 \+ " }{TEXT 312 1 "x" }{TEXT 314 3 " 1 " }{TEXT -1 8 " arrays." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Suppose that we know how to define the determinant of an " }{TEXT 316 2 "n " }{TEXT 315 1 "x" }{TEXT 317 4 " n " }{TEXT -1 70 " array. Let us see how t o use this to define the determinant of an " }{TEXT 321 6 "(n+1) " } {TEXT 320 1 "x" }{TEXT 322 6 " (n+1)" }{TEXT -1 9 " matrix " }{TEXT 318 1 " " }{XPPEDIT 319 1 "A" "I\"AG6\"" }{TEXT -1 45 ". \n\nObserve \+ that by deleting the first row, " }{TEXT 328 1 " " }{XPPEDIT 329 1 "j^ th" ")%\"jG%#thG" }{TEXT -1 11 " column of " }{TEXT 326 1 " " } {XPPEDIT 327 1 "A" "I\"AG6\"" }{TEXT -1 36 " we obtain a square matrix of size " }{TEXT 324 2 "n " }{TEXT 323 1 "x" }{TEXT 325 2 " n" } {TEXT -1 124 ". By assumption we know how to calculate the determinan t of this submatrix. Let us call the determinant of this submatrix " }{TEXT 330 1 " " }{XPPEDIT 331 1 "M[1,j" "&%\"MG6$\"\"\"%\"jG" }{TEXT -1 5 " . \n" }}{PARA 0 "" 0 "" {TEXT -1 13 "The quantity " }{TEXT 353 1 " " }{XPPEDIT 354 1 "(-1)^(1+j)*M[1,j]" "*&),$\"\"\"!\"\",&F%F%% \"jGF%F%&%\"MG6$F%F(F%" }{TEXT -1 21 " is said to be the " }{TEXT 352 8 "cofactor" }{TEXT -1 13 " of entry " }{TEXT 355 1 " " } {XPPEDIT 356 1 "A[1,j]" "&%\"AG6$\"\"\"%\"jG" }{TEXT -1 4 " . " }} {PARA 0 "" 0 "" {TEXT -1 21 "\n\nThe determinant of " }{TEXT 332 1 " \+ " }{XPPEDIT 333 1 "A" "I\"AG6\"" }{TEXT -1 18 " is defined to be" }} {PARA 0 "" 0 "" {TEXT -1 16 " " }{TEXT 334 8 " \+ " }{XPPEDIT 335 1 "det(A) = A[1,1]*M[1,1]-A[1,2]*M[1,2]+A[1,3]*M[1,3]- `...`+(-1)^(N+2)*A[1,N+1]*M[1,N+1]" "/-%$detG6#%\"AG,,*&&F&6$\"\"\"F+F +&%\"MG6$F+F+F+F+*&&F&6$F+\"\"#F+&F-6$F+F2F+!\"\"*&&F&6$F+\"\"$F+&F-6$ F+F9F+F+%$...GF5*(),$F+F5,&%\"NGF+F2F+F+&F&6$F+,&FAF+F+F+F+&F-6$F+,&FA F+F+F+F+F+" }{TEXT -1 3 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " " }} {PARA 0 "" 0 "" {TEXT -1 10 "Thus, if " }{XPPEDIT 336 1 "A=MATRIX([[a ,b],[c,d]])" "/%\"AG-%'MATRIXG6#7$7$%\"aG%\"bG7$%\"cG%\"dG" }{TEXT -1 9 " , then " }{TEXT 337 41 "\n\n \+ " }{XPPEDIT 338 1 "A[1,1]=a,M[1,1]=d" "6$/&%\"AG6$\"\"\"F'%\"aG/&%\" MG6$F'F'%\"dG" }{TEXT -1 8 ", " }{TEXT 339 3 " " }{XPPEDIT 340 1 "A[1,2]=b,M[1,2]=c" "6$/&%\"AG6$\"\"\"\"\"#%\"bG/&%\"MG6$F'F(%\" cG" }{TEXT -1 10 ", and\n \n" }{TEXT 341 2 " " }{XPPEDIT 342 1 "det (A)=a*d-b*c" "/-%$detG6#%\"AG,&*&%\"aG\"\"\"%\"dGF*F**&%\"bGF*%\"cGF*! \"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 343 1 "A=MATRIX([[a,b,c],[d,e,f],[g,h ,i] ])" "/%\"AG-%'MATRIXG6#7%7%%\"aG%\"bG%\"cG7%%\"dG%\"eG%\"fG7%%\"gG %\"hG%\"iG" }{TEXT -1 9 " , then " }{TEXT 344 14 "\n \n " } {XPPEDIT 345 1 "A[1,1]=a,M[1,1]=(e*i-f*h)" "6$/&%\"AG6$\"\"\"F'%\"aG/& %\"MG6$F'F',&*&%\"eGF'%\"iGF'F'*&%\"fGF'%\"hGF'!\"\"" }{TEXT -1 2 ", \+ " }{TEXT 346 7 " " }{XPPEDIT 347 1 "A[1,2]=b,M[1,2]=d*i-f*g" "6$ /&%\"AG6$\"\"\"\"\"#%\"bG/&%\"MG6$F'F(,&*&%\"dGF'%\"iGF'F'*&%\"fGF'%\" gGF'!\"\"" }{TEXT -1 7 ", " }{TEXT 350 2 " " }{XPPEDIT 351 1 "A[ 1,3]=c,M[1,3]=d*h-e*g" "6$/&%\"AG6$\"\"\"\"\"$%\"cG/&%\"MG6$F'F(,&*&% \"dGF'%\"hGF'F'*&%\"eGF'%\"gGF'!\"\"" }{TEXT -1 32 " , \n\nand \n\n \+ " }{TEXT 348 2 " " }{XPPEDIT 349 1 "det(A)=a*(e*i-f *h)-b*(d*i-f*g)+c*(d*h-e*g)" "/-%$detG6#%\"AG,(*&%\"aG\"\"\",&*&%\"eGF *%\"iGF*F**&%\"fGF*%\"hGF*!\"\"F*F**&%\"bGF*,&*&%\"dGF*F.F*F**&F0F*%\" gGF*F2F*F2*&%\"cGF*,&*&F7F*F1F*F**&F-F*F9F*F2F*F*" }{TEXT -1 19 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 " Computational Shortcuts" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "The definition given uses an expa nsion along the first row. It can be shown that we can compute a dete rminant by expanding a" }{TEXT 392 31 "long any row or down any column " }{TEXT -1 12 ". Thus, if " }{XPPEDIT 360 1 "A" "I\"AG6\"" }{TEXT -1 8 " is an " }{TEXT 358 2 "n " }{TEXT 357 1 "x" }{TEXT 359 2 " n" } {TEXT -1 21 " matrix we define " }{TEXT 361 1 " " }{XPPEDIT 362 1 " M[i,j]" "&%\"MG6$%\"iG%\"jG" }{TEXT -1 56 " to be the determinant of \+ the submatrix obtained from " }{XPPEDIT 363 1 "A" "I\"AG6\"" }{TEXT -1 17 " by deleting its " }{TEXT 364 1 " " }{XPPEDIT 365 1 "i^th" ")% \"iG%#thG" }{TEXT -1 8 " row and" }{TEXT 366 1 " " }{XPPEDIT 367 1 "j^ th" ")%\"jG%#thG" }{TEXT -1 46 " column. Such a determinant is said t o be a " }{TEXT 433 5 "minor" }{TEXT -1 6 " of " }{XPPEDIT 432 1 "A " "I\"AG6\"" }{TEXT -1 18 ". The expression " }{TEXT 369 1 " " } {XPPEDIT 370 1 "(-1)^(i+j)*M[i,j]" "*&),$\"\"\"!\"\",&%\"iGF%%\"jGF%F% &%\"MG6$F(F)F%" }{TEXT -1 21 " is said to be the " }{TEXT 368 8 "cof actor" }{TEXT -1 13 " of entry " }{TEXT 371 1 " " }{XPPEDIT 372 1 " A[i,j]" "&%\"AG6$%\"iG%\"jG" }{TEXT -1 25 " . Expanding along the " }{TEXT 373 1 " " }{XPPEDIT 374 1 "i^th" ")%\"iG%#thG" }{TEXT -1 33 " r ow we have \n\n " }{TEXT 375 7 " " }{XPPEDIT 376 1 "det(A)=sum((-1)^(i+j)*A[i,j]*M[i,j],j=1..n) " "/-%$detG6#%\"AG- %$sumG6$*(),$\"\"\"!\"\",&%\"iGF-%\"jGF-F-&F&6$F0F1F-&%\"MG6$F0F1F-/F1 ;F-%\"nG" }{TEXT -1 1 " " }{TEXT 381 1 "." }{TEXT -1 28 " \n\n Expa nding along the " }{TEXT 377 1 " " }{XPPEDIT 378 1 "j^th" ")%\"jG%#th G" }{TEXT -1 33 " row we have \n\n " }{TEXT 379 7 " \+ " }{XPPEDIT 380 1 "det(A)=sum((-1)^(i+j)*A[i,j]*M[i,j],i=1..n) " "/-%$detG6#%\"AG-%$sumG6$*(),$\"\"\"!\"\",&%\"iGF-%\"jGF-F-&F&6$F0F1F- &%\"MG6$F0F1F-/F0;F-%\"nG" }{TEXT -1 1 " " }{TEXT 382 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "Clearly the majo rity of work in calculating a determinant arises in the calculation of the cofactors " }{TEXT 393 1 " " }{XPPEDIT 394 1 "M[i,j]" "&%\"MG6$% \"iG%\"jG" }{TEXT -1 32 " . However, if any matrix entry " }{TEXT 383 1 " " }{XPPEDIT 384 1 "A[i,j]" "&%\"AG6$%\"iG%\"jG" }{TEXT -1 8 " is " }{TEXT 387 1 "0" }{TEXT -1 21 " then the product " }{TEXT 385 1 " " }{XPPEDIT 386 1 "(-1)^(i+j)*A[i,j]*M[i,j]" "*(),$\"\"\"!\"\",&% \"iGF%%\"jGF%F%&%\"AG6$F(F)F%&%\"MG6$F(F)F%" }{TEXT -1 10 " is \+ " }{TEXT 388 1 "0" }{TEXT -1 35 " and we don't need to calculate " }{TEXT 389 1 " " }{XPPEDIT 390 1 "M[i,j]" "&%\"MG6$%\"iG%\"jG" }{TEXT -1 15 " to know this!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "The idea, then, is to exploit whatever zeros are alr eady present in " }{XPPEDIT 391 1 "A" "I\"AG6\"" }{TEXT -1 70 " and to introduce new zeros. We do this by using the following fact:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 395 102 " If we add a multiple of one row to another row then we do not change the determinant of the matrix." }}{PARA 257 "" 0 "" {TEXT 396 108 " If \+ we add a multiple of one column to another column then we do not chang e the determinant of the matrix." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 420 5 "Maple" }{TEXT -1 22 " command for adding " }{TEXT 421 1 "m" }{TEXT -1 13 " times row " }{TEXT 422 2 "i1" }{TEXT -1 13 " of matrix " }{TEXT 424 2 "A " } {TEXT -1 10 " to row " }{TEXT 423 2 "i2" }{TEXT -1 7 " is " } {TEXT 425 20 "addrow(A, i1, i2, m)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 5 "The " }{TEXT 426 5 "Maple" }{TEXT -1 22 " command for \+ adding " }{TEXT 427 1 "m" }{TEXT -1 16 " times column " }{TEXT 428 2 "j1" }{TEXT -1 13 " of matrix " }{TEXT 430 2 "A " }{TEXT -1 13 " \+ to column " }{TEXT 429 3 " j2" }{TEXT -1 7 " is " }{TEXT 431 20 "a ddcol(A, j1, j2, m)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "\n The " }{TEXT 434 5 "Maple" }{TEXT -1 26 " command for deleting the" }{TEXT 440 2 " " }{XPPEDIT 441 1 "i^th" ")%\"iG%#thG" }{TEXT -1 10 " \+ row and " }{TEXT 442 1 " " }{XPPEDIT 443 1 "j^th" ")%\"jG%#thG" } {TEXT -1 20 " column of matrix " }{TEXT 438 2 "A " }{TEXT -1 3 " is " }{TEXT 439 15 " minor(A, i, j)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 446 5 " " }}{PARA 0 "" 0 "" {TEXT 447 8 " " }{TEXT 448 4 "Note" }{TEXT 449 14 ": The command " }{TEXT 444 15 " minor(A, i , j)" }{TEXT -1 2 " " }{TEXT 445 10 "returns a " }{TEXT 468 9 "submat rix" }{TEXT 469 22 ", not its determinant." }}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 8 " Example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "C" }{TEXT 411 8 "alculate" }{TEXT -1 1 " " }{TEXT 397 3 " " }{XPPEDIT 398 1 "det(MATRIX([ [1,2,0,3],[2,3,2,1],[0,1,-10 ,1],[1,1,0,4]])" "-%$detG6#-%'MATRIXG6#7&7&\"\"\"\"\"#\"\"!\"\"$7&F+F- F+F*7&F,F*,$\"#5!\"\"F*7&F*F*F,\"\"%" }{TEXT -1 1 " " }{TEXT 412 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 413 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 5 " \nLet\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "A := matrix([ [1, 2,0,3],[2,3,2,1],[0,1,-10,1],[1,1,0,4]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7&7&\"\"\"\"\"#\"\"!\"\"$7&F+F-F+F*7& F,F*!#5F*7&F*F*F,\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Among the rows and c olumns of " }{XPPEDIT 399 1 "A" "I\"AG6\"" }{TEXT -1 223 ", column th ree has the greatest number of zeros. We therefore expand down column \+ 3. First we introduce one more zero in column 3 by adding five times \+ row 2 to row 3. Doing this does not change the value of the determinan t:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "A1 := addrow(A,2,3,5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G-%'MATRIXG6#7&7&\"\"\"\"\"#\"\"! \"\"$7&F+F-F+F*7&\"#5\"#;F,\"\"'7&F*F*F,\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 6 "\n Thus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 23 " " }{TEXT 401 3 " " }{XPPEDIT 402 1 "det(A) =-2*det(MATRIX([ [1,2,3],[10,16,6],[1,1,4]] ))" "/-%$det G6#%\"AG,$*&\"\"#\"\"\"-F$6#-%'MATRIXG6#7%7%F*F)\"\"$7%\"#5\"#;\"\"'7% F*F*\"\"%F*!\"\"" }{TEXT -1 1 " " }{TEXT 400 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 119 "We can compute this sm aller determinant in the same way: Let's expand along the third row af ter introducing some zeros:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "B \+ := minor(A1, 2, 3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'MATRIX G6#7%7%\"\"\"\"\"#\"\"$7%\"#5\"#;\"\"'7%F*F*\"\"%" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 51 "B1 := addcol(B,1,2,-1); #adds (-1)col[1] to \+ col[2] " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#B1G-%'MATRIXG6#7%7%\"\" \"F*\"\"$7%\"#5\"\"'F.7%F*\"\"!\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "B2 := addcol(B1,1,3,-4); #adds (-4)col[1] to col[3] \+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#B2G-%'MATRIXG6#7%7%\"\"\"F*!\" \"7%\"#5\"\"'!#M7%F*\"\"!F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We can now calculate " }{TEXT 403 1 " " } {XPPEDIT 404 1 "det(B2) = 1*(1*(-34)-(-1)*6)" "/-%$detG6#%#B2G*&\"\"\" F(,&*&F(F(,$\"#M!\"\"F(F(*&,$F(F-F(\"\"'F(F-F(" }{TEXT -1 6 " or " } {TEXT 405 3 "-28" }{TEXT -1 16 " in our head: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 0 "" 0 "" {TEXT -1 16 " " }{TEXT 406 1 " " }{XPPEDIT 407 1 "det(A ) = -2*(-28)" "/-%$detG6#%\"AG,$*&\"\"#\"\"\",$\"#G!\"\"F*F-" }{TEXT -1 2 " " }{XPPEDIT 408 1 "` `=56" "/%\"~G\"#c" }{TEXT -1 1 " " } {TEXT 409 3 ".\n\n" }{TEXT -1 12 "Naturally " }{TEXT 410 5 "Maple" } {TEXT -1 57 " has a simple command with which we can check our work: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#c" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 " Exercises" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 13 "1. Calculate" }{TEXT 414 3 " " } {XPPEDIT 415 1 "det(MATRIX([ [1,2,1,3],[2,3,2,1],[1,1,0,0],[1,2,-1,4]] )" "-%$detG6#-%'MATRIXG6#7&7&\"\"\"\"\"#F*\"\"$7&F+F,F+F*7&F*F*\"\"!F/ 7&F*F+,$F*!\"\"\"\"%" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "2. C alculate" }{TEXT 416 3 " " }{XPPEDIT 417 1 "det(MATRIX([ [1,-2,2,1], [2,3,2,1],[2,1,2,3],[1,2,-1,0]])" "-%$detG6#-%'MATRIXG6#7&7&\"\"\",$\" \"#!\"\"F,F*7&F,\"\"$F,F*7&F,F*F,F/7&F*F,,$F*F-\"\"!" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 13 "3. Calculate" }{TEXT 418 3 " " } {XPPEDIT 419 1 "det(MATRIX([ [3,-3,2,1],[1,1,-1,1],[2,1,2,1],[1,2,-1,2 ]])" "-%$detG6#-%'MATRIXG6#7&7&\"\"$,$F*!\"\"\"\"#\"\"\"7&F.F.,$F.F,F. 7&F-F.F-F.7&F.F-,$F.F,F-" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 " Solutions" }}{PARA 0 "" 0 "" {TEXT -1 39 "1. Expand along the \+ third row to obtain" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "det(matrix([[1, 2, 1, 3], [2, 3, 2, 1], [1, 1, 0, 0], [1, 2, -1, 4]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "2. Exp and across the fourth row or down the fourth column to obtain" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "det(matrix([[1, -2, 2, 1], [2, 3, 2, 1], [2, 1, 2, 3], [1, 2, -1, \+ 0]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#C" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "3. There are no zeros but it is easy to introduce three zeros in the second row to obtain" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "det(matrix([[3, -3, 2, 1], [1, 1, -1, 1], [2, 1, 2, 1], [1, 2, -1, 2]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 " The Wronskian" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 450 5 "Maple" }{TEXT -1 26 " command for creating a " }{TEXT 457 3 " n " }{TEXT 456 1 "x" } {TEXT 458 2 " n" }{TEXT -1 25 " Wronskian matrix is " }{TEXT 455 15 "wronskian(v, x)" }{TEXT -1 8 " where " }{TEXT 459 1 "v" }{TEXT -1 40 " is a list or vector of functions and " }{TEXT 460 1 "x" } {TEXT -1 40 " is the differentiation variable. \n\n" }{TEXT 473 9 "Remember:" }{TEXT -1 2 " " }{TEXT 470 5 "Maple" }{TEXT -1 1 " " } {TEXT 474 1 " " }{TEXT 476 33 "is case-sensitive. The command is" } {TEXT -1 3 " " }{TEXT 471 15 "wronskian(v, x)" }{TEXT -1 3 ", " } {TEXT 475 3 "not" }{TEXT -1 3 " " }{TEXT 472 15 "Wronskian(v, x)" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "expressionLi st := [exp(a*x),exp(b*x),exp(c*x)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%/expressionListG7%-%$expG6#*&%\"aG\"\"\"%\"xGF+-F'6#*&%\"bGF+F,F+-F '6#*&%\"cGF+F,F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "express ionVector := vector([cos(a*x),sin(a*x),cos(b*x),sin(b*x)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1expressionVectorG-%'VECTORG6#7&-%$cosG6#* &%\"aG\"\"\"%\"xGF.-%$sinGF+-F*6#*&%\"bGF.F/F.-F1F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "wronskian(expressionList,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%-%$expG6#*&%\"aG\"\"\"%\"xGF--F )6#*&%\"bGF-F.F--F)6#*&%\"cGF-F.F-7%*&F,F-F(F-*&F2F-F/F-*&F6F-F3F-7%*& F,\"\"#F(F-*&F2F=F/F-*&F6F=F3F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "det( \" );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,.*,-%$expG6#*&% \"aG\"\"\"%\"xGF*F*%\"bGF*-F&6#*&F,F*F+F*F*%\"cG\"\"#-F&6#*&F0F*F+F*F* F**,F%F*F0F*F2F*F,F1F-F*!\"\"*,F)F*F%F*F-F*F0F1F2F*F6*,F)F*F%F*F2F*F,F 1F-F*F**,F)F1F%F*F-F*F0F*F2F*F**,F)F1F%F*F2F*F,F*F-F*F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "factor( \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.-%$expG6#*&%\"bG\"\"\"%\"xGF*F*-F&6#*&%\"cGF*F+F*F* -F&6#*&%\"aGF*F+F*F*,&F/!\"\"F)F*F*,&F3F*F/F5F*,&F3F*F)F5F*F5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "wronskian(expressionVector,x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7&-%$cosG6#*&%\"aG \"\"\"%\"xGF--%$sinGF*-F)6#*&%\"bGF-F.F--F0F27&,$*&F/F-F,F-!\"\"*&F(F- F,F-,$*&F5F-F4F-F9*&F1F-F4F-7&,$*&F(F-F,\"\"#F9,$*&F/F-F,FAF9,$*&F1F-F 4FAF9,$*&F5F-F4FAF97&*&F/F-F,\"\"$,$*&F(F-F,FJF9*&F5F-F4FJ,$*&F1F-F4FJ F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "det( \" );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,:**-%$cosG6#*&%\"aG\"\"\"%\"xGF*\"\"#F)F*-F &6#*&%\"bGF*F+F*F,F0\"\"&F***F%F,F)F*-%$sinGF.F,F0F1F***F%F,F)\"\"$F3F ,F0F6!\"#**F%F,F)F6F-F,F0F6F7**-F4F'F,F)F*F-F,F0F1F***F:F,F)F*F3F,F0F1 F***F:F,F)F6F-F,F0F6F7**F:F,F)F6F3F,F0F6F7**F%F,F)F1F-F,F0F*F***F%F,F) F1F3F,F0F*F***F:F,F)F1F-F,F0F*F***F:F,F)F1F3F,F0F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "factor( \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*.%\"aG\"\"\"%\"bGF%,&*$-%$cosG6#*&F&F%%\"xGF%\"\"#F%*$ -%$sinGF+F.F%F%,&*$-F*6#*&F$F%F-F%F.F%*$-F1F5F.F%F%,&F$F%F&!\"\"F.,&F$ F%F&F%F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 463 3 " " }{TEXT 464 4 "Note" }{TEXT 465 14 ": The command " }{TEXT 461 16 " wronskian(v, x)" }{TEXT -1 2 " " }{TEXT 462 10 "returns a " }{TEXT 466 6 "matrix" }{TEXT 467 22 ", \+ not its determinant." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 33 " Copyright and Author Information" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 484 15 "det_cr ashR4.mws" }{TEXT -1 27 " A MapleV R4 worksheet." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 481 6 "Author" }{TEXT -1 20 ": Brian E. Blank \n" }{TEXT 482 7 "Written" }{TEXT -1 20 ": 6 \+ February 2003\n" }{TEXT 483 12 "Last Revised" }{TEXT -1 20 ": 6 Feb ruary 2003" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "This document may not be distributed by any medium," }}{PARA 0 "" 0 "" {TEXT -1 55 "including print, disk, and electronic transfer, w ithout" }}{PARA 0 "" 0 "" {TEXT -1 39 "prior written permission of the author." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 49 "For more information, please contact the author:" }}{PARA 258 "" 0 "" {TEXT -1 4 " " }}{PARA 258 "" 0 "" {TEXT -1 5 " " } {TEXT 480 25 "Department of Mathematics" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 39 " Washington University in St. Louis" }}{PARA 0 "" 0 "" {TEXT -1 26 " St. Louis, MO 63130" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 33 " Telephone: (31 4) 935-6763" }}{PARA 258 "" 0 "" {TEXT -1 44 " e-mail: \+ brian@math.wustl.edu" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Copyright: \251 2003 Brian E. Blank, All Rights Re served" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}}{MARK "7 0 0" 5 }{VIEWOPTS 1 1 0 1 1 1803 }