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} {CSTYLE "_cstyle22" -1 231 "" 0 1 0 128 128 1 0 0 1 2 2 2 0 0 0 1 } {PSTYLE "_pstyle13" -1 218 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }3 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{CSTYLE "_cstyle23" -1 232 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle24" -1 233 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle14" -1 219 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 206 "" 0 "" {TEXT 210 32 "A Quick Introduction to Linear A" }{TEXT 210 6 "lgebra" }}{PARA 206 "" 0 "" {TEXT 210 20 "for Math 374 at WMU." }}{PARA 207 "" 0 "" {TEXT 211 6 "Part I" }}{PARA 207 "" 0 "" {TEXT 211 11 "Jay Treiman" }}{PARA 208 "" 0 "" }{PARA 207 "" 0 "" {TEXT 211 8 "May 2004" }}{PARA 207 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 53 "The two worksheets in this set only cover the linear " } }{PARA 208 "" 0 "" {TEXT 212 37 "algebra in the syllibus for Math 272 \+ " }{TEXT 212 19 "at WMU. It is not " }}{PARA 208 "" 0 "" {TEXT 212 35 "intended to give any more material " }{TEXT 212 23 "or to give the student " }}{PARA 208 "" 0 "" {TEXT 212 34 "more than a basic introdu ction to " }{TEXT 212 31 "the linear algebra in Math 272." }}}{SECT 1 {PARA 209 "" 0 "" {HYPERLNK 213 "Restart" 2 "restart" "" }{TEXT 214 7 " Maple." }}{PARA 208 "" 0 "" {TEXT 212 47 "This will make certain tha t any old definitions" }}{PARA 208 "" 0 "" {TEXT 212 44 "or setting ar e removed from the kerenl. Any" }}{PARA 208 "" 0 "" {TEXT 212 42 "tex t that is green and uncerlined in this " }}{PARA 208 "" 0 "" {TEXT 212 47 "worksheet is a link. They are all links to the" }}{PARA 208 " " 0 "" {TEXT 212 35 "help on a given command or package." }}{PARA 208 "" 0 "" }{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 215 8 "restart;" }}}} {SECT 1 {PARA 209 "" 0 "" {TEXT 214 20 "Matrices and vectors" }}{PARA 210 "" 0 "" {TEXT 216 57 "A matrix is simply a two dimensional array. \+ Its elements" }}{PARA 208 "" 0 "" {TEXT 212 61 "can be numbers, expre ssions, or functions. The following is " }}{PARA 208 "" 0 "" {TEXT 212 16 "a simple matrix." }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 2 " " }{XPPEDIT 18 0 "A =MATRIX([[1,2,5],[3,4,6]])" "6#/%\" AG-%'MATRIXG6#7$7%\"\"\"\"\"#\"\"&7%\"\"$\"\"%\"\"'" }{TEXT 217 1 " " }{TEXT 212 1 "." }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 4 "T he " }{TEXT 218 16 "size of a matrix" }{TEXT 212 44 " is described by \+ the number of rows and the " }}{PARA 208 "" 0 "" {TEXT 212 31 "number \+ of columns. The matrix " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT 217 1 " \+ " }{TEXT 212 34 " is a 2 by 3 matrix. A matrix is " }}{PARA 208 "" 0 "" {TEXT 218 6 "square" }{TEXT 212 46 " if it has the same number of r ows as columns." }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 16 " The elements of " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT 217 1 " " } {TEXT 212 47 " are labeled by row and column. The first row " }} {PARA 208 "" 0 "" {TEXT 212 9 "of A is " }{XPPEDIT 18 0 "[1,2,5]" "6# 7%\"\"\"\"\"#\"\"&" }{TEXT 217 1 " " }{TEXT 212 26 " and the second co lumn is " }{XPPEDIT 18 0 "MATRIX([[2],[4]])" "6#-%'MATRIXG6#7$7#\"\"#7 #\"\"%" }{TEXT 217 1 " " }{TEXT 212 19 ". This means that " }}{PARA 208 "" 0 "" {TEXT 212 19 "the 1,2 element of " }{XPPEDIT 18 0 "A " "6# %\"AG" }{TEXT 217 1 " " }{TEXT 212 35 " is 2. Sometimes a general mat rix " }{XPPEDIT 18 0 "A " "6#%\"AG" }{TEXT 217 1 " " }{TEXT 212 14 " i s written as" }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 2 " " }{XPPEDIT 18 0 "A = MATRIX([[a[1,1],a[1,2],a[1,3]],[a[2,1],a[2,2],a[2, 3]])" "6#/%\"AG-%'MATRIXG6#7$7%&%\"aG6$\"\"\"F-&F+6$F-\"\"#&F+6$F-\"\" $7%&F+6$F0F-&F+6$F0F0&F+6$F0F3" }{TEXT 217 1 " " }{TEXT 212 3 " = " } {XPPEDIT 18 0 "[a[i,j]]" "6#7#&%\"aG6$%\"iG%\"jG" }{TEXT 217 1 " " } {TEXT 212 1 "." }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 65 "T his enables one to write out matrix operations by specifying the" }} {PARA 208 "" 0 "" {TEXT 212 39 "pattern for all elements of the matrix ." }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 76 "A vector is a \+ one dimensional array that is identified with a column matrix," }} {PARA 208 "" 0 "" {TEXT 212 31 "a matrix with one column. The " } {TEXT 218 21 "dimension of a vector" }{TEXT 212 21 " is simply the num ber" }}{PARA 208 "" 0 "" {TEXT 212 73 "of elements in the vector. The dimension of the vector (1,2,-2,-1) is 4." }}{PARA 208 "" 0 "" } {PARA 208 "" 0 "" {TEXT 212 53 "It is simple to define matrices and ve ctors in Maple." }}{PARA 208 "" 0 "" }{EXCHG {PARA 208 "" 0 "" {TEXT 212 51 "To put more than one line in an execution group use" }}{PARA 208 "" 0 "" {TEXT 212 14 "a shift-enter." }}{PARA 208 "" 0 "" }{PARA 208 "> " 0 "" {MPLTEXT 1 215 27 "A := Matrix([[1,2],[3,4]]);" } {MPLTEXT 1 215 21 "\nA1 := <<1,3>|<2,4>>;" }{MPLTEXT 1 215 22 "\nx := \+ Vector([1,3,4]);" }{MPLTEXT 1 215 15 "\nx1 := <1,3,4>;" }}}{SECT 1 {PARA 210 "" 0 "" {TEXT 216 9 "Exercises" }}{PARA 208 "" 0 "" }{SECT 1 {PARA 211 "" 0 "" {TEXT 219 1 "1" }}{PARA 208 "" 0 "" {TEXT 212 45 " What are the sizes of the following matrices?" }}{PARA 208 "" 0 "" {TEXT 212 2 " " }{XPPEDIT 18 0 "A = MATRIX([[2, 3], [3, -5], [0, 6]]) ;" "6#/%\"AG-%'MATRIXG6#7%7$\"\"#\"\"$7$F+,$\"\"&!\"\"7$\"\"!\"\"'" } {TEXT 217 1 " " }{TEXT 217 4 " , " }{XPPEDIT 18 0 "B = MATRIX([[1, 2, 3, 4], [0, 9, 8, 7]]);" "6#/%\"BG-%'MATRIXG6#7$7&\"\"\"\"\"#\"\"$\"\" %7&\"\"!\"\"*\"\")\"\"(" }{TEXT 217 1 " " }{TEXT 217 8 " , and " } {XPPEDIT 18 0 "C = MATRIX([[3, 0, 1], [0, 1, 0], [1, 1, 1]]);" "6#/%\" CG-%'MATRIXG6#7%7%\"\"$\"\"!\"\"\"7%F+F,F+7%F,F,F," }{TEXT 217 1 " " } {TEXT 217 2 " ." }}{EXCHG {PARA 208 "" 0 "" }}}{SECT 1 {PARA 211 "" 0 "" {TEXT 219 1 "2" }}{PARA 208 "" 0 "" {TEXT 212 49 "What are the dime nsions of the following vectors?" }}{PARA 208 "" 0 "" {TEXT 212 2 " " }{XPPEDIT 18 0 "v = VECTOR([1, 0, 2, 0, 3]);" "6#/%\"vG-%'VECTORG6#7' \"\"\"\"\"!\"\"#F*\"\"$" }{TEXT 217 1 " " }{TEXT 217 3 " , " } {XPPEDIT 18 0 "w = VECTOR([a, b, c, z]);" "6#/%\"wG-%'VECTORG6#7&%\"aG %\"bG%\"cG%\"zG" }{TEXT 217 1 " " }{TEXT 217 9 " , and " }{XPPEDIT 18 0 "z = VECTOR([16, 23, -101]);" "6#/%\"zG-%'VECTORG6#7%\"#;\"#B,$\" $,\"!\"\"" }{TEXT 217 1 " " }{TEXT 217 2 " ." }}{EXCHG {PARA 208 "" 0 "" }}}}}{SECT 1 {PARA 209 "" 0 "" {TEXT 214 24 "Simple Matrix Operatio ns" }}{PARA 208 "" 0 "" {TEXT 212 44 "To do matrix opertaions one shou ld load the " }{HYPERLNK 220 "LinearAlgebra" 2 "LinearAlgebra" "" } {TEXT 212 19 " package for Maple." }}{PARA 208 "" 0 "" {TEXT 212 70 "M any of the linear algebra opertations in Maple will not work if'this " }}{PARA 208 "" 0 "" {TEXT 212 68 "package is not loaded. A colon aft er a command in Maple suppresses " }}{PARA 208 "" 0 "" {TEXT 212 25 "o utput from that command." }}{PARA 208 "" 0 "" }{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 215 20 "with(LinearAlgebra):" }}}{PARA 208 "" 0 "" } {SECT 1 {PARA 210 "" 0 "" {TEXT 216 8 "Addition" }}{PARA 208 "" 0 "" {TEXT 212 68 "One adds matrices or vectors by adding the corresponding elements. " }}{PARA 208 "" 0 "" {TEXT 221 67 "It is assumed that the two matrices being added have the same size." }}{PARA 208 "" 0 "" {TEXT 212 24 "The sum of two matrices " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT 217 1 " " }{TEXT 212 5 " and " }{XPPEDIT 18 0 "B;" "6#%\"BG" } {TEXT 217 1 " " }{TEXT 212 11 " is defined" }{TEXT 212 1 "\n" }}{PARA 208 "" 0 "" {TEXT 212 2 " " }{XPPEDIT 18 0 "A+B = [a[i,j]+b[i,j]]" "6 #/,&%\"AG\"\"\"%\"BGF&7#,&&%\"aG6$%\"iG%\"jGF&&%\"bGF,F&" }{TEXT 217 1 " " }{TEXT 212 1 "." }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 60 "Here is the sum of two arbitray 3 by 2 matri ces. It is done" }}{PARA 208 "" 0 "" {TEXT 212 24 "in two different w ays. " }}{PARA 208 "" 0 "" }{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 215 33 "A := Matrix(2,3,(i,j) -> a[i,j]);" }{MPLTEXT 1 215 34 "\nB := \+ Matrix(2,3,(i,j) -> b[i,j]);" }{MPLTEXT 1 215 22 "\nA_plus_B := Add(A, B);" }{MPLTEXT 1 215 19 "\nA_plus_B := A + B;" }}}{PARA 208 "" 0 "" } {PARA 208 "" 0 "" {TEXT 212 42 "Here is the sum of two specific matric es. " }}{PARA 208 "" 0 "" }{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 215 31 "A := Matrix([[1,2,3],[4,5,6]]);" }{MPLTEXT 1 215 35 "\nB := Matrix ([[7,8,9],[10,11,12]]);" }{MPLTEXT 1 215 19 "\nA_plus_B := A + B;" } {MPLTEXT 1 215 22 "\nA_plus_B := Add(A,B);" }}}{PARA 208 "" 0 "" } {PARA 208 "" 0 "" {TEXT 212 57 "Subtracting one matrix from another is defined as adding " }}{PARA 208 "" 0 "" {TEXT 212 61 "negative one ti mes the second matrix to the first. (See the " }}{PARA 208 "" 0 "" {TEXT 212 14 "next section.)" }}{SECT 1 {PARA 211 "" 0 "" {TEXT 219 9 "Exercises" }}{PARA 208 "" 0 "" {TEXT 212 52 "Add the following pairs \+ of matrices by hand and then" }}{PARA 208 "" 0 "" {TEXT 212 55 "check \+ your answers using Maple. If you do not practice" }}{PARA 208 "" 0 "" {TEXT 212 48 "hand calculations you may have trouble on exams." }} {PARA 208 "" 0 "" }{SECT 1 {PARA 212 "" 0 "" {TEXT 222 1 "3" }}{PARA 208 "" 0 "" {XPPMATH 200 "6#/%\"AG-%'MATRIXG6#7$7$\"\"#\"\"\"7$\"\"$,$ F+!\"\"" }{TEXT 212 6 " and " }{XPPMATH 200 "6#/%\"BG-%'MATRIXG6#7$7$ \"\"%,$\"\"#!\"\"7$\"\"!F," }{TEXT 212 1 "." }}{EXCHG {PARA 208 "> " 0 "" }}}{SECT 1 {PARA 212 "" 0 "" {TEXT 222 1 "4" }}{PARA 208 "" 0 "" {XPPMATH 200 "6#/%\"AG-%'MATRIXG6#7$7%\"\"$,$\"\"'!\"\",$\"\"#F-7%\"\" \",$F*F-\"\"&" }{TEXT 212 6 " and " }{XPPMATH 200 "6#/%\"BG-%'MATRIXG 6#7$7%\"\"',$\"\"#!\"\"\"\"!7%\"\"$,$\"\"%F-\"\"&" }{TEXT 212 1 "." }} {EXCHG {PARA 208 "> " 0 "" }}}{SECT 1 {PARA 212 "" 0 "" {TEXT 222 1 "5 " }}{PARA 208 "" 0 "" {XPPMATH 200 "6#/%\"AG-%'MATRIXG6#7%7$\"\"!\"\" \"7$F+F*7$\"\"%\"\"&" }{TEXT 212 6 " and " }{XPPMATH 200 "6#/%\"BG-%' MATRIXG6#7%7$\"\"\",$\"\"#!\"\"7$F,,$F*F-7$\"\"$\"\"%" }{TEXT 212 1 ". " }}{EXCHG {PARA 208 "> " 0 "" }}}}}{SECT 1 {PARA 210 "" 0 "" {TEXT 216 21 "Scalar Multiplication" }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 58 "Multiplying a matrix or a vector by a scalar is defined \+ as" }}{PARA 208 "" 0 "" {TEXT 212 64 "multiplying every element of the matrix or vector by the scalar," }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 2 " " }{XPPEDIT 18 0 "alpha*A = [alpha*a[i,j]]" "6#/*&%& alphaG\"\"\"%\"AGF&7#*&F%F&&%\"aG6$%\"iG%\"jGF&" }{TEXT 217 1 " " } {TEXT 212 1 "." }}{PARA 208 "" 0 "" }{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 215 9 "print(A);" }{MPLTEXT 1 215 15 "\nMultiply(A,2);" } {MPLTEXT 1 215 5 "\n2*A;" }}}{PARA 208 "" 0 "" }{SECT 1 {PARA 211 "" 0 "" {TEXT 219 9 "Exercises" }}{PARA 208 "" 0 "" {TEXT 212 40 "Do the f ollowing scalar multiplications." }}{SECT 1 {PARA 212 "" 0 "" {TEXT 222 1 "6" }}{PARA 208 "" 0 "" {XPPMATH 200 "6#*&\"\"#\"\"\"-%'MATRIXG6 #7$7$,$F%!\"\"\"\"!7$\"\"$F%F%" }}{EXCHG {PARA 208 "> " 0 "" }}}{SECT 1 {PARA 212 "" 0 "" {TEXT 222 1 "7" }}{PARA 208 "" 0 "" {XPPMATH 200 " 6#,$*&\"\"#\"\"\"-%'MATRIXG6#7$7%,$\"\"'!\"\"\"\"!,$F%F.7%\"\"$F&F2F&F ." }}{EXCHG {PARA 208 "> " 0 "" }}}{SECT 1 {PARA 212 "" 0 "" {TEXT 222 1 "8" }}{PARA 208 "" 0 "" {TEXT 212 1 " " }{XPPMATH 200 "6#*&\"\"% \"\"\"-%'MATRIXG6#7%7$\"#5,$\"#@!\"\"7$\"\"&,$\"\"(F.7$\"\"',$F%F.F%" }}{EXCHG {PARA 208 "> " 0 "" }}}}}{SECT 1 {PARA 210 "" 0 "" {TEXT 216 21 "Matrix multiplication" }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 61 "Matrix multiplication can be viewed as composition of li near " }}{PARA 208 "" 0 "" {TEXT 212 61 "functions. Because of this, \+ the formula is fairly complex. " }}{PARA 208 "" 0 "" {TEXT 212 16 "Th e product of " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT 217 1 " " }{TEXT 212 5 " and " }{XPPEDIT 18 0 "B" "6#%\"BG" }{TEXT 217 1 " " }{TEXT 212 14 " is define if " }{XPPEDIT 18 0 "A " "6#%\"AG" }{TEXT 217 1 " " }{TEXT 212 15 " is m by n and " }{XPPEDIT 18 0 "B" "6#%\"BG" }{TEXT 217 1 " " }{TEXT 212 12 " is n by k. " }}{PARA 208 "" 0 "" {TEXT 212 31 "The result is an m by k matrix " }{XPPEDIT 18 0 "C" "6#%\"CG" } {TEXT 217 1 " " }{TEXT 212 21 " whose i,j element is" }}{PARA 208 "" 0 "" {TEXT 212 1 " " }{XPPEDIT 18 0 "sum(a[i,k]*b[k,j],k=1..n" "6#-%$su mG6$*&&%\"aG6$%\"iG%\"kG\"\"\"&%\"bG6$F+%\"jGF,/F+;F,%\"nG" }{TEXT 217 1 " " }{TEXT 212 1 "." }}{PARA 213 "" 0 "" }{PARA 214 "" 0 "" {TEXT 223 25 "In order for the product " }{XPPEDIT 18 0 "A*B;" "6#*&% \"AG\"\"\"%\"BGF%" }{TEXT 224 1 " " }{TEXT 223 37 " to be defined the \+ number of columns " }}{PARA 215 "" 0 "" {TEXT 225 3 "of " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT 226 1 " " }{TEXT 225 34 " must equal the nu mber of rows of " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT 226 1 " " } {TEXT 225 1 "." }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 41 "F or two arbitrary 2 by 2 matrices one has" }}{PARA 208 "" 0 "" }{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 215 33 "A := Matrix(2,2,(i,j) -> a[i,j] );" }{MPLTEXT 1 215 34 "\nB := Matrix(2,2,(i,j) -> b[i,j]);" } {MPLTEXT 1 215 5 "\nA.B;" }{MPLTEXT 1 215 15 "\nMultiply(A,B);" }}} {PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 82 "Which method one us es for multiplying two matrices is a matter of personal choice." }} {PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 72 "Since a vector is a column matrix, one can mutiply a matrix by a vector." }}{PARA 208 "" 0 "" }{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 215 23 "Y := Vector(2,i->y [i]);" }{MPLTEXT 1 215 5 "\nA.Y;" }}}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 47 "Here is an example of multiplying two specific " } {TEXT 212 9 "matrices." }}{PARA 208 "" 0 "" }{EXCHG {PARA 208 "> " 0 " " {MPLTEXT 1 215 34 "A := Matrix([[2,3,-2],[0,-6,10]]);" }{MPLTEXT 1 215 41 "\nB := Matrix([[2,3,1],[1,0,1],[2,-2,6]]);" }{MPLTEXT 1 215 21 "\nAB := Multiply(A,B);" }}}{SECT 1 {PARA 211 "" 0 "" {TEXT 219 9 " Exercises" }}{PARA 208 "" 0 "" {TEXT 212 57 "Multiply the following pa irs of matrices by hand and then" }}{PARA 208 "" 0 "" {TEXT 212 55 "ch eck your answers using Maple. If you do not practice" }}{PARA 208 "" 0 "" {TEXT 212 48 "hand calculations you may have trouble on exams." } }{PARA 208 "" 0 "" }{SECT 1 {PARA 212 "" 0 "" {TEXT 222 1 "9" }}{PARA 208 "" 0 "" {XPPMATH 200 "6#/%\"AG-%'MATRIXG6#7$7$\"\"#\"\"\"7$\"\"$,$ F+!\"\"" }{TEXT 212 6 " and " }{XPPMATH 200 "6#/%\"BG-%'MATRIXG6#7$7$ \"\"%,$\"\"#!\"\"7$\"\"!F," }{TEXT 212 1 "." }}{EXCHG {PARA 208 "> " 0 "" }}}{SECT 1 {PARA 212 "" 0 "" {TEXT 222 2 "10" }}{PARA 208 "" 0 "" {XPPMATH 200 "6#/%\"AG-%'MATRIXG6#7$7%\"\"$,$\"\"&!\"\",$\"\"#F-7%\"\" \",$F*F-F," }{TEXT 212 6 " and " }{XPPMATH 200 "6#/%\"BG-%'MATRIXG6#7 %7$\"\"',$\"\"#!\"\"7$\"\"$\"\"&7$F,\"\"!" }{TEXT 212 1 "." }}{EXCHG {PARA 208 "> " 0 "" }}}{SECT 1 {PARA 212 "" 0 "" {TEXT 222 2 "11" }} {PARA 208 "" 0 "" {XPPMATH 200 "6#/%\"AG-%'MATRIXG6#7%7$\"\"!\"\"\"7$F +F*7$\"\"%\"\"&" }{TEXT 212 6 " and " }{XPPMATH 200 "6#/%\"BG-%'MATRI XG6#7$7%\"\"\",$\"\"#!\"\"F,7%,$F*F-\"\"$\"\"%" }{TEXT 212 1 "." }} {EXCHG {PARA 208 "> " 0 "" }}}}}}{SECT 1 {PARA 209 "" 0 "" {TEXT 214 16 "Special Matrices" }}{PARA 208 "" 0 "" {TEXT 212 65 "There are seve ral special types of matrices with which you should" }}{PARA 208 "" 0 "" {TEXT 212 65 "be familiar. This is a list of three of those types \+ of matrices." }}{SECT 1 {PARA 210 "" 0 "" {TEXT 216 19 "The identity m atrix" }}{PARA 208 "" 0 "" {TEXT 212 64 "An identity matrix is a squar e matrix with each diagonal element" }}{PARA 208 "" 0 "" {TEXT 212 68 "equal to 1 and all other elements equal to 0. The diagonal elements" }}{PARA 208 "" 0 "" {TEXT 212 19 "of an n x n matrix " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT 217 1 " " }{TEXT 212 18 " are the elements " } {XPPEDIT 18 0 "a[i,i];" "6#&%\"aG6$%\"iGF&" }{TEXT 217 1 " " }{TEXT 212 20 ". Here is a 3 by 3 " }}{PARA 208 "" 0 "" {TEXT 212 16 "identi ty matrix." }}{PARA 208 "" 0 "" {TEXT 212 2 " " }{XPPEDIT 18 0 "I[3] \+ = MATRIX([[1, 0, 0], [0, 1, 0], [0, 0, 1]]);" "6#/&%\"IG6#\"\"$-%'MATR IXG6#7%7%\"\"\"\"\"!F.7%F.F-F.7%F.F.F-" }{TEXT 217 1 " " }{TEXT 212 1 "." }}{PARA 208 "" 0 "" {TEXT 212 51 "To get an identity matrix in Map le one can use the " }{HYPERLNK 220 "diag" 2 "linalg,diag" "" }{TEXT 212 9 " command." }}{PARA 208 "" 0 "" }{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 215 29 "Ident_4 := IdentityMatrix(4);" }}}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 26 "The product of any matrix " } {XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT 217 1 " " }{TEXT 212 28 " with an \+ appropriately sized" }}{PARA 208 "" 0 "" {TEXT 212 26 "identity matrix is always " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT 217 1 " " }{TEXT 212 1 "." }}{PARA 208 "" 0 "" }{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 215 40 "A := Matrix([[2,4,-1,3],[9,100,-23,7]]);" }{MPLTEXT 1 215 30 " \nIdent_2 := IdentityMatrix(2);" }{MPLTEXT 1 215 21 "\nMultiply(A,Iden t_4);" }{MPLTEXT 1 215 21 "\nMultiply(Ident_2,A);" }}}}{SECT 1 {PARA 210 "" 0 "" {TEXT 216 15 "The zero matrix" }}{PARA 208 "" 0 "" {TEXT 212 56 "A zero matrix is a matrix of any size whose elements are" }} {PARA 208 "" 0 "" {TEXT 212 9 "all zero." }}{PARA 208 "" 0 "" {TEXT 212 2 " " }{XPPEDIT 18 0 "Z = MATRIX([[0, 0, 0, 0, 0], [0, 0, 0, 0, 0 ], [0, 0, 0, 0, 0]]);" "6#/%\"ZG-%'MATRIXG6#7%7'\"\"!F*F*F*F*F)F)" } {TEXT 217 1 " " }{TEXT 212 2 " ." }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 46 "If the sizes match, the product of any matrix " } {XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT 217 1 " " }{TEXT 212 22 " with a z ero matrix is" }}{PARA 208 "" 0 "" {TEXT 212 36 "always zero. The sum of any matrix " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT 217 1 " " } {TEXT 212 22 " and a zero matrix is " }{XPPEDIT 18 0 "B;" "6#%\"BG" } {TEXT 217 1 " " }{TEXT 212 1 "." }}{PARA 208 "" 0 "" }{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 215 9 "print(B);" }{MPLTEXT 1 215 22 "\nZ := \+ ZeroMatrix(3,3);" }{MPLTEXT 1 215 18 "\nZ_times_B := Z.B;" }{MPLTEXT 1 215 17 "\nZ_plus_B := Z+B;" }}}}{SECT 1 {PARA 210 "" 0 "" {TEXT 216 17 "Diagonal matrices" }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 61 "A square matrix is diagonal if all the elements of the matrix" }}{PARA 208 "" 0 "" {TEXT 212 61 "off the diagonal are zero. Here is a 3 by 3 diagonal matrix." }}{PARA 216 "" 0 "" {TEXT 227 2 " " } {XPPEDIT 18 0 "D = MATRIX([[1, 0, 0], [0, 2, 0], [0, 0, 3]]);" "6#/%\" DG-%'MATRIXG6#7%7%\"\"\"\"\"!F+7%F+\"\"#F+7%F+F+\"\"$" }{TEXT 228 1 " \+ " }{TEXT 227 1 "." }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 55 "It is easy to generate a diagonal matrix in Maple using" }}{PARA 208 "" 0 "" {TEXT 212 4 "the " }{HYPERLNK 220 "diag" 2 "linalg,diag" " " }{TEXT 212 9 " command." }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 215 32 "Diag := DiagonalMatrix([1,2,3]);" }}}}}{SECT 1 {PARA 209 "" 0 "" {TEXT 214 31 "Properties of matrix operations" }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 58 "The operations of matrix addition and multiplication have " }}{PARA 208 "" 0 "" {TEXT 212 56 "similar prope rties to the same scalar operations. There" }}{PARA 208 "" 0 "" {TEXT 212 16 "are differences." }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 39 "Here is a list of properties that hold." }}{PARA 208 "" 0 "" }{SECT 1 {PARA 210 "" 0 "" {TEXT 216 32 "Properties of matrix op erations." }}{PARA 208 "" 0 "" {TEXT 212 17 "In the following " } {XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT 217 1 " " }{TEXT 212 2 ", " } {XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT 217 1 " " }{TEXT 212 6 ", and " } {XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT 217 1 " " }{TEXT 212 18 " are matr ices and " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 217 1 " " } {TEXT 212 5 " and " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT 217 1 " \+ " }}{PARA 208 "" 0 "" {TEXT 212 50 "are scalars. The following are al ways true if the" }}{PARA 208 "" 0 "" {TEXT 212 23 "operations are def ined." }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 6 "(1) " } {XPPEDIT 18 0 "A+B = B+A;" "6#/,&%\"AG\"\"\"%\"BGF&,&F'F&F%F&" }{TEXT 217 1 " " }}{PARA 208 "" 0 "" {TEXT 212 6 "(2) " }{XPPEDIT 18 0 "A+B +C = A+B+C;" "6#/,(%\"AG\"\"\"%\"BGF&%\"CGF&F$" }{TEXT 217 1 " " }} {PARA 208 "" 0 "" {TEXT 212 6 "(3) " }{XPPEDIT 18 0 "0+A = A;" "6#/, &\"\"!\"\"\"%\"AGF&F'" }{TEXT 217 1 " " }}{PARA 208 "" 0 "" {TEXT 212 6 "(4) " }{XPPEDIT 18 0 "alpha*(A+B) = alpha*A+alpha*B;" "6#/*&%&alp haG\"\"\",&%\"AGF&%\"BGF&F&,&*&F%F&F(F&F&*&F%F&F)F&F&" }{TEXT 217 1 " \+ " }}{PARA 208 "" 0 "" {TEXT 212 6 "(5) " }{XPPEDIT 18 0 "(alpha+beta )*A = alpha*A+alpha*B;" "6#/*&,&%&alphaG\"\"\"%%betaGF'F'%\"AGF',&*&F& F'F)F'F'*&F&F'%\"BGF'F'" }{TEXT 217 1 " " }}{PARA 208 "" 0 "" {TEXT 212 7 "(6) (" }{XPPEDIT 18 0 "alpha*beta;" "6#*&%&alphaG\"\"\"%%beta GF%" }{TEXT 217 1 " " }{TEXT 212 1 ")" }{XPPEDIT 18 0 "A = alpha;" "6# /%\"AG%&alphaG" }{TEXT 217 1 " " }{TEXT 212 1 "(" }{XPPEDIT 18 0 "beta *A;" "6#*&%%betaG\"\"\"%\"AGF%" }{TEXT 217 1 " " }{TEXT 212 4 ") = " } {XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT 217 1 " " }{TEXT 212 1 "(" } {XPPEDIT 18 0 "alpha*A;" "6#*&%&alphaG\"\"\"%\"AGF%" }{TEXT 217 1 " " }{TEXT 212 1 ")" }}{PARA 208 "" 0 "" {TEXT 212 6 "(7) " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT 217 1 " " }{TEXT 212 1 " " }{XPPEDIT 18 0 " -A = 0;" "6#/,$%\"AG!\"\"\"\"!" }{TEXT 217 1 " " }}{PARA 208 "" 0 "" {TEXT 212 6 "(8) " }{XPPEDIT 18 0 "I*A = A;" "6#/*&%\"IG\"\"\"%\"AGF &F'" }{TEXT 217 1 " " }}{PARA 208 "" 0 "" {TEXT 212 6 "(9) " } {XPPEDIT 18 0 "A*(B+C) = A*B+A*C;" "6#/*&%\"AG\"\"\",&%\"BGF&%\"CGF&F& ,&*&F%F&F(F&F&*&F%F&F)F&F&" }{TEXT 217 1 " " }}{PARA 208 "" 0 "" {TEXT 212 6 "(10) " }{XPPEDIT 18 0 "(B+C)*A = B*A+C*A;" "6#/*&,&%\"BG \"\"\"%\"CGF'F'%\"AGF',&*&F&F'F)F'F'*&F(F'F)F'F'" }{TEXT 217 1 " " }} {PARA 208 "" 0 "" {TEXT 212 6 "(11) " }{XPPEDIT 18 0 "A;" "6#%\"AG" } {TEXT 217 1 " " }{TEXT 212 1 "(" }{XPPEDIT 18 0 "BC;" "6#%#BCG" } {TEXT 217 1 " " }{TEXT 212 5 ") = (" }{XPPEDIT 18 0 "AB;" "6#%#ABG" } {TEXT 217 1 " " }{TEXT 212 1 ")" }{XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT 217 1 " " }}{PARA 208 "" 0 "" {TEXT 212 6 "(12) " }{XPPEDIT 18 0 "alp ha;" "6#%&alphaG" }{TEXT 217 1 " " }{TEXT 212 1 "(" }{XPPEDIT 18 0 "A* B;" "6#*&%\"AG\"\"\"%\"BGF%" }{TEXT 217 1 " " }{TEXT 212 5 ") = (" } {XPPEDIT 18 0 "alpha*A;" "6#*&%&alphaG\"\"\"%\"AGF%" }{TEXT 217 1 " " }{TEXT 212 1 ")" }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT 217 1 " " } {TEXT 212 3 " = " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT 217 1 " " } {TEXT 212 1 "(" }{XPPEDIT 18 0 "alpha*B;" "6#*&%&alphaG\"\"\"%\"BGF%" }{TEXT 217 1 " " }{TEXT 212 1 ")" }}}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 69 "It is important to note that the order of multiplica tion matters even" }}{PARA 208 "" 0 "" {TEXT 212 30 "when both matrice s are square." }}{PARA 208 "" 0 "" }{SECT 1 {PARA 210 "" 0 "" {TEXT 216 39 "Matrix multiplication does not commute." }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 215 41 "A := Matrix([[1,0,1],[2,2,2],[1,-1,-1]]);" }{MPLTEXT 1 215 42 "\nB := Matrix([[2,0,2],[1,-1,-1],[3,4,3]]);" } {MPLTEXT 1 215 18 "\nA_times_B := A.B;" }{MPLTEXT 1 215 18 "\nB_times_ A := B.A;" }}}}{PARA 208 "" 0 "" }{SECT 1 {PARA 210 "" 0 "" {TEXT 216 9 "Exercises" }}{EXCHG {PARA 208 "" 0 "" {TEXT 212 9 "Multiply " } {XPPEDIT 18 0 "A = MATRIX([[1, 2, 3], [1, 2, 3], [1, 2, 3]]);" "6#/%\" AG-%'MATRIXG6#7%7%\"\"\"\"\"#\"\"$F)F)" }{TEXT 217 1 " " }{TEXT 212 24 " by the diagonal matrix " }{XPPEDIT 18 0 "Diag1 = MATRIX([[5, 0, 0 ], [0, 2, 0], [0, 0, 4]]);" "6#/%&Diag1G-%'MATRIXG6#7%7%\"\"&\"\"!F+7% F+\"\"#F+7%F+F+\"\"%" }{TEXT 217 1 " " }}{PARA 208 "" 0 "" {TEXT 212 38 "on both sides. Note that multiplying " }{XPPEDIT 18 0 "A;" "6#%\" AG" }{TEXT 217 1 " " }{TEXT 212 43 " on the right multiplies the colum ns by the" }}{PARA 208 "" 0 "" {TEXT 212 72 "diagonal elements and mul tiplying on the left multiplies the rows by the" }}{PARA 208 "" 0 "" {TEXT 212 18 "diagonal elements." }}}{SECT 1 {PARA 211 "" 0 "" {TEXT 219 2 "12" }}{PARA 208 "" 0 "" {TEXT 212 9 "Multiply " }{XPPEDIT 18 0 "A = MATRIX([[1, 2, 3], [1, 2, 3], [1, 2, 3]]);" "6#/%\"AG-%'MATRIXG6# 7%7%\"\"\"\"\"#\"\"$F)F)" }{TEXT 217 1 " " }{TEXT 217 24 " by the diag onal matrix " }{XPPEDIT 18 0 "Diag1 = MATRIX([[5, 0, 0], [0, 2, 0], [0 , 0, 4]]);" "6#/%&Diag1G-%'MATRIXG6#7%7%\"\"&\"\"!F+7%F+\"\"#F+7%F+F+ \"\"%" }{TEXT 217 1 " " }}{PARA 208 "" 0 "" {TEXT 212 38 "on both side s. Note that multiplying " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT 217 1 " " }{TEXT 217 43 " on the right multiplies the columns by the" }} {PARA 208 "" 0 "" {TEXT 212 72 "diagonal elements and multiplying on t he left multiplies the rows by the" }}{PARA 208 "" 0 "" {TEXT 212 18 " diagonal elements." }}{EXCHG {PARA 208 "> " 0 "" }}}{SECT 1 {PARA 211 "" 0 "" {TEXT 219 2 "13" }}{PARA 208 "" 0 "" {TEXT 212 8 "Use the " } {HYPERLNK 220 "randmatrix" 2 "linalg,randmatrix" "" }{TEXT 212 59 " fu nction in Maple to generate 10 pairs of 3 by 3 matrices " }}{PARA 208 "" 0 "" {XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT 217 1 " " }{TEXT 217 5 " a nd " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT 217 1 " " }{TEXT 217 65 " wh ose entries are integers between -100 and 100. Check if they " }} {PARA 208 "" 0 "" {TEXT 212 14 "commute, does " }{XPPEDIT 18 0 "A*B = \+ B*A;" "6#/*&%\"AG\"\"\"%\"BGF&*&F'F&F%F&" }{TEXT 217 1 " " }{TEXT 217 1 "?" }}{EXCHG {PARA 208 "> " 0 "" }}}}}{SECT 1 {PARA 209 "" 0 "" {TEXT 214 38 "Two more ways of getting new matrices." }}{SECT 1 {PARA 210 "" 0 "" {TEXT 216 14 "The transpose." }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 4 "The " }{TEXT 218 9 "transpose" }{TEXT 212 13 " of a matrix " }{XPPEDIT 18 0 "A = [a[i,j]];" "6#/%\"AG7#&%\"aG6$%\"i G%\"jG" }{TEXT 217 1 " " }{TEXT 212 14 " is the matrix" }}{PARA 208 "" 0 "" {XPPEDIT 18 0 "A^T = [a[j,i]];" "6#/)%\"AG%\"TG7#&%\"aG6$%\"jG% \"iG" }{TEXT 217 1 " " }{TEXT 212 16 ". As an example" }}{PARA 217 "" 0 "" {TEXT 229 1 " " }{XPPEDIT 18 0 "MATRIX([[1, 2, 3], [4, 5, 6]])^T = MATRIX([[1, 4], [2, 5], [3, 6]]);" "6#/)-%'MATRIXG6#7$7%\"\"\"\"\"# \"\"$7%\"\"%\"\"&\"\"'%\"TG-F&6#7%7$F*F.7$F+F/7$F,F0" }{TEXT 230 1 " " }{TEXT 229 1 "." }}{PARA 208 "" 0 "" {TEXT 212 45 "One can view this \+ as exchanging the rows and " }}{PARA 208 "" 0 "" {TEXT 212 22 "columns of a matrix. " }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 22 " In maple one uses the " }{HYPERLNK 231 "Transpose" 2 "LinearAlgebra,Tr anspose" "" }{TEXT 212 1 " " }{TEXT 212 7 "command" }{TEXT 212 1 "." } }{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 215 31 "A := Matrix([[1,2,3],[4 ,5,6]]);" }{MPLTEXT 1 215 14 "\nTranspose(A);" }}}{PARA 208 "" 0 "" } {PARA 208 "" 0 "" {TEXT 212 19 "A square matrix is " }{TEXT 218 9 "sym metric" }{TEXT 212 4 " if " }{XPPEDIT 18 0 "A^T = A;" "6#/)%\"AG%\"TGF %" }{TEXT 217 1 " " }{TEXT 212 13 ". Here is a " }}{PARA 208 "" 0 "" {TEXT 212 18 "symmetric matrix ." }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 2 " " }{XPPEDIT 18 0 "S = MATRIX([[1, 2, 3], [2, 3, 4], \+ [3, 4, 5]]);" "6#/%\"SG-%'MATRIXG6#7%7%\"\"\"\"\"#\"\"$7%F+F,\"\"%7%F, F.\"\"&" }{TEXT 217 1 " " }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 41 "There are rules for computing transposes." }}{SECT 1 {PARA 211 "" 0 "" {TEXT 219 36 "Rlues for computing with transposes." }}{PARA 208 "" 0 "" {TEXT 212 6 "(1) " }{XPPEDIT 18 0 "(A^T)^T = A;" "6#/))%\"AG%\"TGF'F&" }{TEXT 217 1 " " }}{PARA 208 "" 0 "" {TEXT 212 6 "(2) " }{XPPEDIT 18 0 "(A+B)^T = A^T+B^T;" "6#/),&%\"AG\"\"\"%\"BG F'%\"TG,&)F&F)F')F(F)F'" }{TEXT 217 1 " " }}{PARA 208 "" 0 "" {TEXT 212 6 "(3) " }{XPPEDIT 18 0 "(A*B)^T = B^T*A^T;" "6#/)*&%\"AG\"\"\"% \"BGF'%\"TG*&)F(F)F')F&F)F'" }{TEXT 217 1 " " }}{PARA 208 "" 0 "" } {PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 51 "It is easy to check if the rules are equalities for" }}{PARA 208 "" 0 "" {TEXT 212 25 "2 \+ by 2 matrices in Maple." }}{PARA 208 "" 0 "" }{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 215 31 "A := Matrix(2,2,(i,j)->a[i,j]);" }{MPLTEXT 1 215 34 "\nB := Matrix(2,2,(i,j) -> b[i,j]);" }{MPLTEXT 1 215 73 "\nTra nspose(Multiply(A,B)) + ((-1)*(Multiply(Transpose(B),Transpose(A))));" }}}}{PARA 208 "" 0 "" }}{SECT 1 {PARA 210 "" 0 "" {TEXT 216 24 "The i nverse of a matrix." }}{PARA 208 "" 0 "" {TEXT 212 31 "The inverse of \+ a square matrix " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT 217 1 " " } {TEXT 212 13 " is a matrix " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT 217 1 " " }{TEXT 212 10 " such that" }}{PARA 218 "" 0 "" {TEXT 232 2 " " }{XPPEDIT 18 0 "AB = I;" "6#/%#ABG%\"IG" }{TEXT 233 1 " " }{TEXT 232 5 " and " }{XPPEDIT 18 0 "BA = I;" "6#/%#BAG%\"IG" }{TEXT 233 1 " " } {TEXT 232 1 "." }}{PARA 208 "" 0 "" {TEXT 212 69 "It is important to n ote that not all square matrices have an inverse." }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 13 "For example, " }{XPPEDIT 18 0 "C = M ATRIX([[0, 1], [0, 0]]);" "6#/%\"CG-%'MATRIXG6#7$7$\"\"!\"\"\"7$F*F*" }{TEXT 217 1 " " }{TEXT 212 48 " does not have an inverse. This is ea sy to show" }}{PARA 208 "" 0 "" {TEXT 212 24 "since the first row of \+ " }{XPPEDIT 18 0 "B*C;" "6#*&%\"BG\"\"\"%\"CGF%" }{TEXT 217 1 " " } {TEXT 212 41 " is a linear combination of the rows of " }{XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT 217 1 " " }{TEXT 212 5 ". It" }}{PARA 208 "" 0 "" {TEXT 212 28 "cannot be the first row of " }{XPPEDIT 18 0 "I;" " 6#%\"IG" }{TEXT 217 1 " " }{TEXT 212 47 " since there would be a 1 in the 1,1 position." }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" }{SECT 1 {PARA 211 "" 0 "" {TEXT 219 18 "Computing inverses" }}{PARA 208 "" 0 " " }{PARA 208 "" 0 "" {TEXT 212 72 "When using matrices in practice one should not compute the inverse of a " }}{PARA 208 "" 0 "" {TEXT 212 75 "matrix unless it is essential. There are a number of ways to calc ulate an " }}{PARA 208 "" 0 "" {TEXT 212 56 "inverse. You can look th em up in a linear algebra text." }}{PARA 208 "" 0 "" }{PARA 208 "" 0 " " {TEXT 212 35 "With Maple one can use the command " }{HYPERLNK 220 "i nverse" 2 "linalg,inverse" "" }{TEXT 212 34 " to find the inverse of a matrix. " }}{PARA 208 "" 0 "" }{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 215 29 "A := Matrix([[3,-2],[4,-1]]);" }{MPLTEXT 1 215 18 "\nMatrixInv erse(A);" }}}{EXCHG {PARA 208 "> " 0 "" }}}{PARA 208 "" 0 "" }{SECT 1 {PARA 211 "" 0 "" {TEXT 219 33 "Rules for computing with inverses" }} {PARA 208 "" 0 "" {TEXT 212 22 "The following hold if " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT 217 1 " " }{TEXT 212 30 " and B are invertable matrixs." }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 212 6 "(1) " }{XPPEDIT 18 0 "(A^(-1))^(-1);" "6#))%\"AG,$\"\"\"!\"\"F&" }{TEXT 217 1 " " }}{PARA 208 "" 0 "" {TEXT 212 6 "(2) " }{XPPEDIT 18 0 "(A* B)^(-1) = B^(-1)*A^(-1);" "6#/)*&%\"AG\"\"\"%\"BGF',$F'!\"\"*&)F(F)F') F&F)F'" }{TEXT 217 1 " " }}}}{SECT 1 {PARA 210 "" 0 "" {TEXT 216 9 "Ex ercises" }}{PARA 208 "" 0 "" }{SECT 1 {PARA 211 "" 0 "" {TEXT 219 2 "1 4" }}{PARA 208 "" 0 "" {TEXT 212 35 "Find the transposes of the matric es" }}{SECT 1 {PARA 212 "" 0 "" {TEXT 222 1 "a" }}{PARA 208 "" 0 "" {TEXT 212 2 " " }{XPPMATH 200 "6#/%\"AG-%'MATRIXG6#7$7%\"\"\"\"\"#\" \"$7%\"\"%\"\"&\"\"'" }}{EXCHG {PARA 208 "> " 0 "" }}}{SECT 1 {PARA 212 "" 0 "" {TEXT 222 1 "b" }}{PARA 208 "" 0 "" {TEXT 212 1 " " } {XPPMATH 200 "6#/%\"BG-%'MATRIXG6#7%7%\"\"\"\"\"#F*7%F+\"\"$,$F*!\"\"7 %F*F.\"\"&" }}{EXCHG {PARA 208 "> " 0 "" }}}{SECT 1 {PARA 212 "" 0 "" {TEXT 222 1 "c" }}{PARA 208 "" 0 "" {TEXT 212 1 " " }{XPPMATH 200 "6#/ %\"CG-%'MATRIXG6#7%7#\"\"\"7#\"\"#7#\"\"$" }}{EXCHG {PARA 208 "> " 0 " " }}}}{SECT 1 {PARA 211 "" 0 "" {TEXT 219 2 "15" }}{PARA 208 "" 0 "" {TEXT 212 54 "Check if the following pairs of matrices are inverses." }}{SECT 1 {PARA 212 "" 0 "" {TEXT 222 1 "a" }}{PARA 208 "" 0 "" {TEXT 212 3 " " }{XPPMATH 200 "6#/%\"AG-%'MATRIXG6#7$7$\"\"#\"\"\"7$F+F*" }{TEXT 212 7 " and " }{XPPMATH 200 "6#/%\"BG-%'MATRIXG6#7$7$*&\"\"# \"\"\"\"\"$!\"\",$*&F,F,F-F.F.7$F/F*" }{TEXT 212 1 " " }}{EXCHG {PARA 208 "> " 0 "" }}}{SECT 1 {PARA 212 "" 0 "" {TEXT 222 1 "b" }}{PARA 208 "" 0 "" {TEXT 212 2 " " }{XPPMATH 200 "6#/%\"AG-%'MATRIXG6#7&7&\" \"#\"\"!F+F+7&F+\"\"%F+F+7&F+F+\"\"$F+7&F+F+F+\"\"&" }{TEXT 212 7 " a nd " }{XPPMATH 200 "6#/%\"BG-%'MATRIXG6#7&7&*&\"\"\"F+\"\"#!\"\"\"\"! F.F.7&F.*&F+F+\"\"%F-F.F.7&F.F.*&F+F+\"\"$F-F.7&F.F.F.*&F+F+\"\"&F-" } }{EXCHG {PARA 208 "> " 0 "" }}}{SECT 1 {PARA 212 "" 0 "" {TEXT 222 1 " c" }}{PARA 208 "" 0 "" {TEXT 212 1 " " }{XPPMATH 200 "6#/%\"AG-%'MATRI XG6#7%7%\"\"\"\"\"#\"\"!7%F+\"\"$F*7%F*F*F*" }{TEXT 212 7 " and " } {XPPMATH 200 "6#/%\"BG-%'MATRIXG6#7%7%,$*&\"\"$\"\"\"\"\"#!\"\"F/,$F-F /*&F-F-F.F/7%F1\"\"!,$F1F/7%F+F0F1" }}{EXCHG {PARA 208 "> " 0 "" }}}} {SECT 1 {PARA 211 "" 0 "" {TEXT 219 2 "16" }}{PARA 208 "" 0 "" {TEXT 212 43 "Use Maple to find the inverse of the matrix" }}{PARA 208 "" 0 "" {TEXT 212 2 " " }{XPPEDIT 18 0 "C = MATRIX([[2, 3], [1, 2]]);" "6# /%\"CG-%'MATRIXG6#7$7$\"\"#\"\"$7$\"\"\"F*" }{TEXT 217 1 " " }}{EXCHG {PARA 208 "> " 0 "" }}}}}{PARA 219 "" 0 "" }}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }