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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 20 "More on Determinants" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "A := matrix(2,2,[a,b,c,d]);\n" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "A^`-1` " "6#)%\"AG%#-1G" }{TEXT 
-1 2 "= " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "inverse(A);\n" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Check det(inverse(A)) = 1/det(A
)" }{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "d
et(inverse(A));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "\n" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "What about determ
inants of square matrices of order more than  2?" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 2 "\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 40 "A := matrix(3,3,[1,2,3,0,-2,2,3,7,-4]);\n" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 14 "minor(A,1,1);\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "minor(A,2,3);\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "minor(A,3,1);\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "mi := (i,j) -> det(minor(A,i,j));\n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 35 "co := (i,j) -> (-1)^(i+j)*mi(i,j);\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "M := matrix(3,3,mi);\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "C := matrix(3,3,co);\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Exercise: Given the matrices A and
 B below, find the relationship between det(A) and det(B)." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "A := matrix(3,3,[a,b,c,p,q,r,x,y,z]
);\nB:= matrix(3,3,[-x,-y,-z,3*p+a,3*q+b,3*r+c,2*p,2*q,2*r]);\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "det(A); " }{TEXT -1 0 "" }
{MPLTEXT 1 0 7 "det(B);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Exerci
se: Find the relationship among the determinants of the following matr
ices." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "A := matrix(3,3,[1
,-1,3,1,0,-1,2,1,6]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "
A1 := addrow(A,1,2,-1);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
25 "A2 := addrow(A1,1,3,-2);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 25 "A3 := addrow(A2,2,3,-3);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 91 "None of these operations affected the determinant,  so  det A  \+
= det A3 = (1)(1)(12)  = 12." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 54 "A := matrix(4,4,[1,0,3,1,2,2,6,0,4,0,-3,1,4,1,12,1]);\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "The idea is to reduce to triangula
r form using elementary row operations." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "A1 := addrow(A,1,2,-2);\n" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 25 "A2 := addrow(A1,1,3,-4);\n" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 24 "A3 :=addrow(A2,1,4,-4);\n" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "
A4 := mulrow(A3,2,1/2);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A5 := addrow(A4,2,4,-1
);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 55 "A := matrix(4,4,[4,-1,3,-1,0,1,2,2,3,1,0,2,
1,2,-1,1]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A1 :=swapr
ow(A,1,4);\n" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "det A = - det A
1" "6#/*&%$detG\"\"\"%\"AGF&,$*&F%F&%#A1GF&!\"\"" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 25 "A2 := addrow(A1,1,3,-3);\n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 25 "A3 := addrow(A2,1,4,-4);\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A4 := addrow(A3,2,3,5);\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A5 := addrow(A4,2,4,9);\n" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "det A1 = det A2 = det A3 = det A
4 = det A5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A6 := mulrow(
A5,3,1/13);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "A7 := addr
ow(A6,3,4,-25);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 41 "A := matrix(3,3,[2,4,-9,3,-1,2,-3,5,6]);\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "A1 := matrix(3,3,[3,-1,2,2,4
,-9,-3,5,6]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "A2 := ma
trix(3,3,[-1,3,2,4,2,-9,5,-3,6]);\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "A3 := matrix(3,3,[-1,3,2,0,14,-1,0,12,16]);\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "A4 := matrix(3,3,[-1,2,3,0,-
1,14,0,16,12]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A5 := \+
matrix(3,3,[-1,2,3,0,-1,14,0,0,12+16*(14)]);\n" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "d
et(A);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "A := matrix(3,3,[1,4,1,2,-1,0,0,18,4]);\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A1 := addrow(A,1,2,-2);\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "det A = det A1 = 0 (row 3 is (-2) \+
times row 2)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Note. If  A and \+
B are nxn matrices   then   " }{XPPEDIT 18 0 "det(AB) = (det A)(det B)
" "6#/-%$detG6#%#ABG-*&F%\"\"\"%\"AGF*6#*&F%F*%\"BGF*" }{TEXT -1 1 ".
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Example:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 36 "A := matrix(3,3); B := matrix(3,3);\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A=evalm(A); B=evalm(B);\n" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "det(A&*B);\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "det(A);\n" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 8 "det(B);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "det(A)*det(B);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "d
et(A)*det(B)-det(A&*B);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "simplify(%);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "A := \+
matrix(2,2,[1,2,3,4]);   B:= matrix(2,2,[-1,2,5,6]);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 35 "detA := det(A);    detB := det(B);\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "AB := evalm(A&*B);   detAB:=
 det(AB);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "det (3*A);\n" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 16 "This is  NOT    " }{XPPEDIT 18 0 "3* det(A) " "6#*&
\"\"$\"\"\"-%$detG6#%\"AGF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
58 "A := matrix(2,2,[1,2,3,4]);   B:= matrix(2,2,[-1,2,5,6]);\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "`A+B` :=evalm(A+B);\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "det(`A+B`) := det(A+B);\n" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "detA :=det(A); detB := det
(B);\n" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "det A + det B " "6#,&
*&%$detG\"\"\"%\"AGF&F&*&F%F&%\"BGF&F&" }{TEXT -1 5 "  =  " }{XPPEDIT 
18 0 " -18 <> det(A+B)" "6#0,$\"#=!\"\"-%$detG6#,&%\"AG\"\"\"%\"BGF," 
}}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 28 "Eigenvalues and Eigenvectors" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 8 "Example." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 71 "A := matrix(3,3,[0.9167,0.0833,0,0.0833,0.9167,0,-0.1
25,-0.125,1.25]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eval
m(lambda&*id - A);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "cha
rpoly :=charpoly(A,lambda);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 18 "factor(charpoly);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "eigenvalues(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eige
nvects(A);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Example." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "A := matrix(2,2,[1,4,2,3]);\n" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 " lambda I - A" "6#,&*&%'lambdaG
\"\"\"%\"IGF&F&%\"AG!\"\"" }{TEXT -1 3 " = " }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 23 "evalm(lambda&*id - A);\n" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 31 "charpoly :=charpoly(A,lambda);\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "factor(charpoly);\n" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 42 "To find the eigenvectors corresponding to
 " }{XPPEDIT 18 0 "lambda = 5" "6#/%'lambdaG\"\"&" }{TEXT -1 34 " , we
 solve the homogeneous system" }{XPPEDIT 18 0 " (lambda I - A)*x = 0 \+
" "6#/*&,&*&%'lambdaG\"\"\"%\"IGF(F(%\"AG!\"\"F(%\"xGF(\"\"!" }{TEXT 
-1 8 "  with  " }{XPPEDIT 18 0 "lambda = 5.;" "6#/%'lambdaG-%&FloatG6$
\"\"&\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eigenvects(A)
;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Example." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 29 "A :=  matrix([[1,2],[3,1]]);\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eigenvects(A);\n" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 8 "Example." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "A := matrix([ [1.6,2.1,5.9],[2.1,3.2,4.3],[5.9,4.3,-1
.3] ]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eigenvects(A);
\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "102 0 0" 
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