{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 26 "Properties of Determinants" }} {PARA 0 "" 0 "" {TEXT 273 0 "" }}{PARA 4 "" 0 "" {TEXT -1 24 "Linear A lgebra Project " }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 256 12 " Objectives:" }}{PARA 0 "" 0 "" {TEXT -1 6 " Use " }{TEXT 257 6 " Maple " }}{PARA 0 "" 0 "" {TEXT -1 41 " 1. To create random square m atrices. " }}{PARA 0 "" 0 "" {TEXT -1 43 " 2. To study properties o f determinants. " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 258 16 " Solved Problem:" }}{PARA 0 "" 0 "" {TEXT -1 113 " 1. Construct \+ a square matrix, find its transpose and verify that determinants of bo th the matrices are equal. " }}{PARA 0 "" 0 "" {TEXT -1 97 " 2. Find a relation between the inverse of a square matrix with the determinan t of the matrix. " }}{PARA 0 "" 0 "" {TEXT -1 80 " 3. Study the effe ct of the following on a determinant and write a conclusion:" }}{PARA 0 "" 0 "" {TEXT -1 58 " a) Multiply a row of the determinant by \+ a constant." }}{PARA 0 "" 0 "" {TEXT -1 48 " b) Interchange two \+ rows of a determinant." }}{PARA 0 "" 0 "" {TEXT -1 69 " c) Const ruct a determinant with a row consisting of all zeros." }}{PARA 0 "" 0 "" {TEXT -1 58 " d) Construct a determinant with two identical rows." }}{PARA 0 "" 0 "" {TEXT -1 48 " e) Add a multiple of one row to another. " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 259 1 " " }{TEXT 260 9 "Solution:" }}{PARA 4 "" 0 "" {TEXT 261 4 "1. \+ " }{TEXT 262 58 "A random square matrix is created by randmatrix opera tion." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }} {PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra ce have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A := randmatrix(3,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"#j\"#d!#f7%\"#X!\")!#$*7%\"##*\"#V!#i" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ATr := transpose(A);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ATrG-%'matrixG6#7%7%\"#j\"#X\"##*7% \"#d!\")\"#V7%!#f!#$*!#i" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " det(A); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!'mI?" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 9 "det(ATr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# !'mI?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "This shows that the dete rminants of a matrix and its transpose have the same value." }}{PARA 0 "" 0 "" {TEXT -1 164 "An important consequence is that the propertie s for the rows of a determinant apply to columns by merely substitutin g the word column/s for row/s in the properties." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 2 "2." }{TEXT -1 219 " In t his part we show that the inverse of a matrix is the adjoint of the ma trix divided by its determinant. The adjoint A is formed by taking th e values of the minors of the entries along with the sign for that ent ry." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "inverse(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'mat rixG6#7%7%#!%&\\%\"'mI?#!$(**F*#\"%tdF*7%#\"%$)G\"'L:5#!$h(F2#!%-;F27% #!%rEF*#!%NDF*#\"%pIF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "a djoint(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"%&\\% \"$(**!%td7%!%md\"%A:\"%/K7%\"%rE\"%ND!%pI" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "adjoint(A)/det(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"'mI?F&-%'matrixG6#7%7%\"%&\\%\"$(**!%td7%!%md\"%A:\"%/ K7%\"%rE\"%ND!%pIF&!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 63 "From the above we see that the inverse(A) = adjoint(A)/det(A)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 6 "3. a)" }{TEXT -1 216 " We multiply t he first row of the matrix A by c and find the determinant of the resu lting matrix. We copy and paste the matrix A from the Maple output in \+ part 1, multiply by the first row by 3 and call the matrix Ac." }} {PARA 0 "" 0 "" {TEXT -1 6 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "A := matrix([[63, 57, -59], [45, -8, -93], [92, 43, - 62]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"#j\" #d!#f7%\"#X!\")!#$*7%\"##*\"#V!#i" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Ac:= matrix( [[189, 171, -177], [45, -8, -93], [92, 4 3, -62]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AcG-%'matrixG6#7%7%\" $*=\"$r\"!$x\"7%\"#X!\")!#$*7%\"##*\"#V!#i" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "det(Ac);" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!')>4'" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "3*det(A);" }}{PARA 11 "" 1 " " {TEXT -1 0 "" }{XPPMATH 20 "6#!')>4'" }}}{PARA 0 "" 0 "" {TEXT 276 127 "Conclusion: If a row of a determinant is multiplied by a constan t, the value of the determinant is multiplied by the constant." }} {PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 265 2 "b)" }{TEXT -1 99 " We copy and paste A, and interchange the first and second rows and the v alue of the determinants." }}{PARA 0 "" 0 "" {TEXT -1 7 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "A21 := matrix([ [45, -8, -93 ],[63, 57, -59], [92, 43, -62]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %$A21G-%'matrixG6#7%7%\"#X!\")!#$*7%\"#j\"#d!#f7%\"##*\"#V!#i" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "det(A21); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"'mI?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 266 94 "Conclusion: If two rows are interchanged the sig n of the value of the determinant is changed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 267 4 "c) " }{TEXT -1 82 "We consider the matrix A with last row substituted by ze ros and call the matrix B." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "B := matrix([[63, 57, -59], [45, -8 , -93], [0, 0, 0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matri xG6#7%7%\"#j\"#d!#f7%\"#X!\")!#$*7%\"\"!F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"! " }}}{PARA 258 "" 1 "" {TEXT -1 0 "" }{TEXT 272 84 "Conclusion: If a \+ row of a determinant is zero the value of the determinant is zero." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " " } {TEXT 268 4 "d) " }{TEXT -1 73 "We take the matrix A and make the fir st rows equal and call the matrix C." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "C := matrix([[-85, -55, -37], [-85, -55, -37], [79, \+ 56, 49]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'matrixG6#7%7%!# &)!#b!#PF)7%\"#z\"#c\"#\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT 269 112 "Conclusion: If the row of a determinant consists of t wo identical rows, the value of the determinant is zero. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 270 4 " e) " }{TEXT -1 121 " We multiply 3 times of first row of A with the s econd row, call the resulting matrix E and compute the determinant of \+ E." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "E := matrix([[63, 57, -59], 3*[63, 57, -59]+[45, -8, \+ -93], [92, 43, -62]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EG-%'mat rixG6#7%7%\"#j\"#d!#f7%\"$M#\"$j\"!$q#7%\"##*\"#V!#i" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(E);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #!'mI?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 271 12 "Conclusion: " }{TEXT -1 1 " " }{TEXT 277 126 "If a multiple of a row is added to ano ther row, the value of the new determinant equals the value of the ori ginal determinant." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 3 "" 0 " " {TEXT -1 39 "_______________________________________" }}{PARA 256 " " 0 "" {TEXT -1 10 "ASSIGNMENT" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 274 11 "Problem 1: " }} {PARA 0 "" 0 "" {TEXT -1 19 "Use the matrix A = " }{XPPEDIT 18 0 "RTAB LE(7382420,MATRIX([[a, b, c], [d, e, f], [g, h, i]]));" "6#-%'RTABLEG6 $\"(?CQ(-%'MATRIXG6#7%7%%\"aG%\"bG%\"cG7%%\"dG%\"eG%\"fG7%%\"gG%\"hG% \"iG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "a) Find the tran spose of A. What is the relation between the determinants of A and tr anspose of A?" }}{PARA 0 "" 0 "" {TEXT -1 75 "b) Find the inverse of \+ A, adjoint of A and determinant of A and show that " }}{PARA 0 "" 0 " " {TEXT -1 41 " inverse(A) = adjoint(A)/det(A)." }}{PARA 0 " " 0 "" {TEXT -1 48 "c) Describe the effect on the determinant of A " }}{PARA 0 "" 0 "" {TEXT -1 55 " i) if a column of A is multiplie d by a constant." }}{PARA 0 "" 0 "" {TEXT -1 42 " ii) if two colu mns are interchanged." }}{PARA 0 "" 0 "" {TEXT -1 46 " iii) if one column consists of all zeros." }}{PARA 0 "" 0 "" {TEXT -1 38 " iv ) if two columns are identical." }}{PARA 0 "" 0 "" {TEXT -1 61 " \+ v) if a multiple of a column is added to a column of A." }}}{PARA 258 "" 1 "" {TEXT -1 1 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 275 10 "Problem 2:" }}{PARA 0 "" 0 "" {TEXT -1 79 "Generate a random 4 x 4 matrix and answer all the questions posed in Problem 1." }}} {PARA 0 "" 0 "" {TEXT -1 76 "_________________________________________ ___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 172 "M SIP Grant #P120A80089-98: \"Three Urban Calculus Reform programs: Ado pting the Best\" 1998-2001, MSEIP Grant #P120A010031: \"Four Colleges: Calculus + Enhancements\" 2001-04" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{MARK "5 45 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 1 33 1 1 }