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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "A := Matrix([[3,3],[4,5
]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\")+8T7-%'MA
TRIXG6#7$7$\"\"$F.7$\"\"%\"\"&%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "Determinant(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"
\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Is det(5A) = 5 det(A)?" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "e:=5;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"eG\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
4 "e.A;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")?:A7-%'MATRI
XG6#7$7$\"#:F,7$\"#?\"#D%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "Determinant(e.A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
\"#v" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Answer to the question \"
Is det(5A) = 5 det(A)?\":_______________" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 146 "If k is a constant and M is a 2 by 2 matrix, is det(kM) \+
= k det(M) true under some condition on k? Use the results of the next
 four command lines." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "M :
= Matrix([[a,b],[c,d]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'R
TABLEG6%\"(#HEL-%'MATRIXG6#7$7$%\"aG%\"bG7$%\"cG%\"dG%'MatrixG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Determinant(M);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#,&*&%\"aG\"\"\"%\"dGF&F&*&%\"bGF&%\"cGF&!\"\"" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "k*Determinant(M);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(\"\"#\"\"\"%\"aGF&%\"dGF&F&*(F%F&%
\"bGF&%\"cGF&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Deter
minant(k.M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(\"\"%\"\"\"%\"aGF&
%\"dGF&F&*(F%F&%\"bGF&%\"cGF&!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 55 "By equating the two preceeding outputs, it must be that" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 8 "k = k^2.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "What \+
number(s) can k equal?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Use the
 same method to answer this question:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 256 "" 0 "" {TEXT -1 47 "If k is a constant and M is a 3 by \+
3 matrix, is" }}{PARA 256 "" 0 "" {TEXT -1 50 "det(kM) = k det(M) true
 under some condition on k?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 98 "Show individual command line entries for M, Det
erminant(M), k*Determinant(M), and Determinant(kM)." }}{PARA 0 "" 0 "
" {TEXT -1 31 "Find all allowable values of k." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 60 "N := Matrix([[0,1,2,-1],[2,5,-7,3],[0,3,6,2],[
-2,-5,4,-2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG-%'RTABLEG6%\"
)7<K7-%'MATRIXG6#7&7&\"\"!\"\"\"\"\"#!\"\"7&F0\"\"&!\"(\"\"$7&F.F5\"\"
'F07&!\"#!\"&\"\"%F9%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "Determinant(N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!#I" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 157 "Since the determinant is not zero
, the inverse matrix exists. One way to find the inverse of a matrix i
s to use the adjoint (classical adjoint or adjugate). " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Adjoint(N);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'RTABLEG6%\")s<Y8-%'MATRIXG6#7&7&\"#a\"#q\"\"(\"#&)7&
!#C!#?!\"#F27&\"\"'\"#5F3F67&\"#=\"\"!!\"'F9%'MatrixG" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 56 "The following formula should compute the \+
inverse matrix." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "(1/Deter
minant(N))*Adjoint(N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%
\")'z#o9-%'MATRIXG6#7&7&#!\"*\"\"&#!\"(\"\"$#F0\"#I#!#<\"\"'7&#\"\"%F.
#\"\"#F1#\"\"\"\"#:F:7&#!\"\"F.#FAF1F<FB7&#!\"$F.\"\"!#F=F.FF%'MatrixG
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Compare this result to the Li
nearAlgebra package's direct computation of the inverse matrix:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "MatrixInverse(N);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")K!Q[\"-%'MATRIXG6#7&7&#!\"*\"
\"&#!\"(\"\"$#F0\"#I#!#<\"\"'7&#\"\"%F.#\"\"#F1#\"\"\"\"#:F:7&#!\"\"F.
#FAF1F<FB7&#!\"$F.\"\"!#F=F.FF%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 214 "How does the LinearAlgebra package compute the inverse o
f a matrix? Do you think it uses the adjoint? See the text section on \+
Cramer's rule. If the package does not use the adjoint, then what meth
od is more likely?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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