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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 6 "Pivots" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "A:=randmatrix(4,7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"AG-%'matrixG6#7&7)!#E!#i\"\"\"!#Z!#\"*F-!#h7)\"#T!#e!#!*\"#`!\"
\"\"#%*\"#$)7)!#')\"#B!#%)\"#>!#]\"#))!#`7)\"#&)\"#\\\"#y\"#<\"#s!#**!
#&)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "pivot(A,1,1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7)!#E!#i\"\"\"!#Z!#\"*F+
!#h7)\"\"!#!%D?\"#8#!%*H#\"#E#!$\\&F5#!$*G\"\"##\"$<&F5#!$V$F57)F/#\"%
lHF2#!%N6F2#\"%oAF2\"$^##\"%lJF2#\"%M>F27)F/#!%)*>F2#\"%8@F5#!%`NF5#!$
^%F:#!%plF5#!%&R(F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "pivo
t(%,2,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7)!#E\"\"!#
\"&%Ht\"%D?#!%%o)\"$D##!&3y'F,#!'-76F,#!'#*G6F,7)F)#!%D?\"#8#!%*H#\"#E
#!$\\&F<#!$*G\"\"##\"$<&F<#!$V$F<7)F)F)#!'*ev\"\"$5)#\"&>H\"\"#!*#\"&L
>$FI#\"'(y?#FI#\"'d[5FI7)F)F)#\"&x_#\"$]\"#!%\"z&\"#]#!&RC\"FV#!&T3%FV
#!&62%FV" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "pivot(%,3,3);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7)!#E\"\"!F)#!)#)G%G\"
\"'Xz()#!)i$>O#F,#!(/6l\"\"'*ev\"#!)1%o*HF,7)F)#!%D?\"#8F)#!*&p_==\"(d
EG##!*?4bm$F:#!*+\\S3#F:#!*Nmk]\"F:7)F)F)#!'*ev\"\"$5)#\"&>H\"\"#!*#\"
&L>$FD#\"'(y?#FD#\"'d[5FD7)F)F)F)#!(@vr$F,#!)r$**e%F,#!)\\Eg5F1#!*86$*
\\\"F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "pivot(%,4,4);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7)!#E\"\"!F)F)#\"*;$>9d
\"(@vr$#\"*1$R0uF,#\"+_uMm?F,7)F)#!%D?\"#8F)F)#\",DA?w(R\")txK[#\",v)o
?\\]F7#\"-+$z+4_\"F77)F)F)#!'*ev\"\"$5)F)#!.CP3F!4E\"+0gf0:#!.%=]8crEF
B#!.j+<;9_)FB7)F)F)F)#!(@vr$\"'Xz()#!)r$**e%FJ#!)\\Eg5\"'*ev\"#!*86$*
\\\"FJ" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Modify " }{TEXT 257 6 "p
ivot " }{TEXT -1 47 "to divide pivot row by pivot. Recall the three " 
}{TEXT 258 25 "elementary row operations" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 71 "E_1 := proc(n, i, j)\n   swaprow(array(identity, 1..n
,1..n), i, j);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "E_2
 := proc(n, i, alpha)\n   mulrow(array(identity, 1..n,1..n), i, alpha)
;\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "E_3 := proc(n,
 i, j, alpha)\n   addrow(array(identity, 1..n,1..n), i, j, alpha);\nen
d:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "The modified pivot, written
 as " }{TEXT 259 5 "Pivot" }{TEXT -1 4 ", is" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 73 "Pivot:=proc(A,i,j) linalg[pivot](E_2(rowdim(A),i
,1/A[i,j]) &* A,i,j) end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&PivotG
R6%%\"AG%\"iG%\"jG6\"F*F*-&%'linalgG6#%&pivotG6%-%#&*G6$-%$E_2G6%-%'ro
wdimG6#9$9%*&\"\"\"F=&F:6$F;9&!\"\"F:F;F@F*F*F*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 13 "Pivot(A,2,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%'matrixG6#7&7)#!%*H#\"#!*#!%>G\"#X\"\"!#!%xTF*#!%\">)F*#!%o?F-#!%
2aF*7)#!#TF*#\"#HF-\"\"\"#!#`F*#F<F*#!#ZF-#!#$)F*7)#!%k=\"#:#\"%d6FGF.
#!$d%FG#!$O(FG#\"\"%FG#!%d>FG7)#\"%3=FG#!#>FGF.#\"$W*FG#\"%n5FG#!$j#FG
#!$'>FG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "v:=vector(36,[])
: for k from 1 to 36 do v[k]:=k od:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "A:=matrix(6,6,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"AG-%'matrixG6#7(7(\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'7(\"\"(\"\")\"\"*
\"#5\"#6\"#77(\"#8\"#9\"#:\"#;\"#<\"#=7(\"#>\"#?\"#@\"#A\"#B\"#C7(\"#D
\"#E\"#F\"#G\"#H\"#I7(\"#J\"#K\"#L\"#M\"#N\"#O" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 13 "Pivot(A,1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%'matrixG6#7(7(\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'7(\"\"!!\"'!#7!#=!#C
!#I7(F/F1F3!#O!#[!#g7(F/F2F6!#a!#s!#!*7(F/F3F7F;!#'*!$?\"7(F/F4F8F<F?!
$]\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Pivot(%,2,2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7(7(\"\"\"\"\"!!\"\"!\"#!
\"$!\"%7(F)F(\"\"#\"\"$\"\"%\"\"&7(F)F)F)F)F)F)F3F3F3" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rref(A);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7(7(\"\"\"\"\"!!\"\"!\"#!\"$!\"%7(F)F(\"\"#
\"\"$\"\"%\"\"&7(F)F)F)F)F)F)F3F3F3" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "X:=randmatrix(6,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"XG-%'matrixG6#7(7(!\"&!#G\"\"%!#6\"#5\"#d7(!##)!#[F-\"#Q!\"(\"#e
7(!#%*!#o\"#9!#N!#9!\"*7(!#^!#tF?!#\"*\"\"\"\"\"&7(!#')\"#V!\"%!#]\"#]
\"#n7(!#R\"\")!#\\\"#6\"#$*F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "rref(X &* A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7(7
(\"\"\"\"\"!!\"\"!\"#!\"$!\"%7(F)F(\"\"#\"\"$\"\"%\"\"&7(F)F)F)F)F)F)F
3F3F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 285 3 "Why" }{TEXT -1 2 " ?" }}
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