{VERSION 5 0 "IBM INTEL NT" "5.0" }
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{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 "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 
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0 1 }}
{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "p:=x^8-21*x^7+186*x^
6-906*x^5+2649*x^4-4749*x^3+5084*x^2-2964*x+720;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"pG,4*$)%\"xG\"\")\"\"\"F**&\"#@F*)F(\"\"(F*!\"\"*&
\"$'=F*)F(\"\"'F*F**&\"$1*F*)F(\"\"&F*F/*&\"%\\EF*)F(\"\"%F*F**&\"%\\Z
F*)F(\"\"$F*F/*&\"%%3&F*)F(\"\"#F*F**&\"%kHF*F(F*F/\"$?(F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "s:=gcd(p,diff(p,x));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"sG,**$)%\"xG\"\"$\"\"\"F**&\"\"'F*)F(\"\"#F*
!\"\"*&\"#6F*F(F*F*F,F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "
d:=gcd(s,diff(s,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG\"\"\"" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "1 is obviously a root of  's'" 
}{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "quo(s,
x-1,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"#\"\"\"F(*&\"
\"&F(F&F(!\"\"\"\"'F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "The othe
r two roots are then  2  and  3. All of these are doublee roots of  'p
'." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "qu
o(p,(x-1)^2*(x-2)^2*(x-3)^2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*
$)%\"xG\"\"#\"\"\"F(*&\"\"*F(F&F(!\"\"\"#?F(" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 42 "The remaining two roots are thus  4 and 5." }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "expand((x-1)^2*(
x-2)^2*(x-3)^2*(x-4)*(x-5)); # to verify" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,4*$)%\"xG\"\")\"\"\"F(*&\"#@F()F&\"\"(F(!\"\"*&\"$'=F()F&\"\"'F
(F(*&\"$1*F()F&\"\"&F(F-*&\"%\\EF()F&\"\"%F(F(*&\"%\\ZF()F&\"\"$F(F-*&
\"%%3&F()F&\"\"#F(F(*&\"%kHF(F&F(F-\"$?(F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 7 "SET 6.3" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 3 "Q14" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning
, the protected names norm and trace have been redefined and unprotect
ed\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Asave:=matrix(3,4,[
2,-2,4,0,-3,3,-6,5,1,-1,2,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&A
saveG-%'matrixG6#7%7&\"\"#!\"#\"\"%\"\"!7&!\"$\"\"$!\"'\"\"&7&\"\"\"!
\"\"F*F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "A:=augment(Asav
e,[0,15,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7'
\"\"#!\"#\"\"%\"\"!F-7'!\"$\"\"$!\"'\"\"&\"#:7'\"\"\"!\"\"F*F-F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "A:=mulrow(A,1,1/2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7'\"\"\"!\"\"\"\"#
\"\"!F-7'!\"$\"\"$!\"'\"\"&\"#:F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "A:=addrow(A,1,2,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"AG-%'matrixG6#7%7'\"\"\"!\"\"\"\"#\"\"!F-7'F-F-F-\"\"&\"#:F)" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "A:=addrow(A,1,3,-1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7'\"\"\"!\"\"\"\"#
\"\"!F-7'F-F-F-\"\"&\"#:7'F-F-F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "A:=mulrow(A,2,1/5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"AG-%'matrixG6#7%7'\"\"\"!\"\"\"\"#\"\"!F-7'F-F-F-F*\"\"$7'F-F-F-
F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "[0,0,0,3]+[1,1,0,0]
*c1+[-2,0,1,0]*c2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(7&\"\"!F%F%\"
\"$\"\"\"*&7&F'F'F%F%F'%#c1GF'F'*&7&!\"#F%F'F%F'%#c2GF'F'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "evalm(Asave &* %);   #just checking
!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"!\"#:F'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Q16:" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Asave:=matrix(4,4,[2,3,1,-11,5,-2,5
,-4,1,-1,3,-3,3,4,-7,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&AsaveG
-%'matrixG6#7&7&\"\"#\"\"$\"\"\"!#67&\"\"&!\"#F/!\"%7&F,!\"\"F+!\"$7&F
+\"\"%!\"(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "A:=augment(
Asave,[1,5,3,-7]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG
6#7&7'\"\"#\"\"$\"\"\"!#6F,7'\"\"&!\"#F/!\"%F/7'F,!\"\"F+!\"$F+7'F+\"
\"%!\"(F*F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "A:=mulrow(A,
1,1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7'\"\"
\"#\"\"$\"\"##F*F-#!#6F-F.7'\"\"&!\"#F2!\"%F27'F*!\"\"F,!\"$F,7'F,\"\"
%!\"(F-F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "for i from 2 t
o 4 do A:=addrow(A,1,i,-A[i,1]) end do;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"AG-%'matrixG6#7&7'\"\"\"#\"\"$\"\"##F*F-#!#6F-F.7'\"\"!#!#>F
-#\"\"&F-#\"#ZF-F57'F*!\"\"F,!\"$F,7'F,\"\"%!\"(F-F>" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7'\"\"\"#\"\"$\"\"##F*F-#!#6F-F.
7'\"\"!#!#>F-#\"\"&F-#\"#ZF-F57'F2#!\"&F-F5F5F57'F,\"\"%!\"(F-F>" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7'\"\"\"#\"\"$\"\"
##F*F-#!#6F-F.7'\"\"!#!#>F-#\"\"&F-#\"#ZF-F57'F2#!\"&F-F5F5F57'F2#!\"
\"F-#!#<F-#\"#PF-F?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A:=m
ulrow(A,2,-2/19);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6
#7&7'\"\"\"#\"\"$\"\"##F*F-#!#6F-F.7'\"\"!F*#!\"&\"#>#!#ZF5F37'F2#F4F-
#\"\"&F-F:F:7'F2#!\"\"F-#!#<F-#\"#PF-F?" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "for i from 3 to 4 do A:=addrow(A,2,i,-A[i,2]) end do;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7'\"\"\"#\"\"$
\"\"##F*F-#!#6F-F.7'\"\"!F*#!\"&\"#>#!#ZF5F37'F2F2#\"#NF5#!#qF5F97'F2#
!\"\"F-#!#<F-#\"#PF-F@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'mat
rixG6#7&7'\"\"\"#\"\"$\"\"##F*F-#!#6F-F.7'\"\"!F*#!\"&\"#>#!#ZF5F37'F2
F2#\"#NF5#!#qF5F97'F2F2#!$k\"F5#\"$G$F5F>" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 25 "A:=addrow(A,2,1,-A[1,2]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7'\"\"\"\"\"!#\"#<\"#>#!#MF.F,7'F+F
*#!\"&F.#!#ZF.F27'F+F+#\"#NF.#!#qF.F77'F+F+#!$k\"F.#\"$G$F.F<" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A:=mulrow(A,3,19/35);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7'\"\"\"\"\"!#\"#<
\"#>#!#MF.F,7'F+F*#!\"&F.#!#ZF.F27'F+F+F*!\"#F*7'F+F+#!$k\"F.#\"$G$F.F
9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "for i to 2 do A:=addro
w(A,3,i,-A[i,3]) end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'m
atrixG6#7&7'\"\"\"\"\"!F+F+F+7'F+F*#!\"&\"#>#!#ZF/F-7'F+F+F*!\"#F*7'F+
F+#!$k\"F/#\"$G$F/F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matri
xG6#7&7'\"\"\"\"\"!F+F+F+7'F+F*F+!\"$F+7'F+F+F*!\"#F*7'F+F+#!$k\"\"#>#
\"$G$F3F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A:=addrow(A,3,
4,-A[4,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7'
\"\"\"\"\"!F+F+F+7'F+F*F+!\"$F+7'F+F+F*!\"#F*7'F+F+F+F+F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "[0,0,1,0]+[0,3,2,1]*c;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#,&7&\"\"!F%\"\"\"F%F&*&7&F%\"\"$\"\"#F&F&%\"cGF
&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalm(Asave &* %);  \+
 #to verify" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7&\"\"\"\"
\"&\"\"$!\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "SET 6.7, Q8:" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "A:=matr
ix(3,6,[1,2,3,1,0,0,4,5,6,0,1,0,7,8,9,0,0,01]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7(\"\"\"\"\"#\"\"$F*\"\"!F-7(\"\"%
\"\"&\"\"'F-F*F-7(\"\"(\"\")\"\"*F-F-F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "for i from 2 to 3 do A:=addrow(A,1,i,-A[i,1]) end do;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7(\"\"\"\"\"#
\"\"$F*\"\"!F-7(F-!\"$!\"'!\"%F*F-7(\"\"(\"\")\"\"*F-F-F*" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7(\"\"\"\"\"#\"\"$F*\"\"!F-
7(F-!\"$!\"'!\"%F*F-7(F-F0!#7!\"(F-F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "A:=mulrow(A,2,-1/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"AG-%'matrixG6#7%7(\"\"\"\"\"#\"\"$F*\"\"!F-7(F-F*F+#\"\"%F,#!\"
\"F,F-7(F-!\"'!#7!\"(F-F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "A:=addrow(A,2,1,-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'m
atrixG6#7%7(\"\"\"\"\"!!\"\"#!\"&\"\"$#\"\"#F/F+7(F+F*F1#\"\"%F/#F,F/F
+7(F+!\"'!#7!\"(F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "A:=
addrow(A,2,3,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#
7%7(\"\"\"\"\"!!\"\"#!\"&\"\"$#\"\"#F/F+7(F+F*F1#\"\"%F/#F,F/F+7(F+F+F
+F*!\"#F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "The matrix is singul
ar (no inverse)!!!" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 8 "SET 7.1:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 4 "Q10:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "Asave:=matrix(3,3,[-10,10,-15,10,5,-30,-5,-10,0]);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&AsaveG-%'matrixG6#7%7%!#5\"#5!#:7%
F+\"\"&!#I7%!\"&F*\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "
A:=evalm(Asave-L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG
6#7%7%,&\"#5!\"\"%\"LGF,F+!#:7%F+,&\"\"&\"\"\"F-F,!#I7%!\"&!#5,$F-F," 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A:=swaprow(A,3,1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%!\"&!#5,$%\"LG!
\"\"7%\"#5,&\"\"&\"\"\"F-F.!#I7%,&F0F.F-F.F0!#:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 60 "for i from 2 to 3 do A:=addrow(A,1,i,-A[i,1]/A
[1,1]) end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%
7%!\"&!#5,$%\"LG!\"\"7%\"\"!,&\"#:F.F-F.,&*&\"\"#\"\"\"F-F6F.\"#IF.7%,
&\"#5F.F-F.F:!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#
7%7%!\"&!#5,$%\"LG!\"\"7%\"\"!,&\"#:F.F-F.,&*&\"\"#\"\"\"F-F6F.\"#IF.7
%F0,&F7F6*&F5F6F-F6F6,&*&,&F5F.*&\"\"&F.F-F6F.F6F-F6F.F2F." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "A:=addrow(A,2,3,2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%!\"&!#5,$%\"LG!\"\"7%\"\"!,&
\"#:F.F-F.,&*&\"\"#\"\"\"F-F6F.\"#IF.7%F0F0,(*&\"\"%F6F-F6F.\"#vF.*&,&
F5F.*&\"\"&F.F-F6F.F6F-F6F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "mul(A[i,i],i=1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"&\"
\"\",&\"#:!\"\"%\"LGF)F&,(*&\"\"%F&F*F&F)\"#vF)*&,&\"\"#F)*&F%F)F*F&F)
F&F*F&F)F&F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "p:=expand(%
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,**&\"$D&\"\"\"%\"LGF(!\"
\"\"%DcF**&\"\"&F()F)\"\"#F(F(*$)F)\"\"$F(F(" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 18 "gcd(p,diff(p,L));;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,&%\"LG\"\"\"\"#:F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "mean
s that  -15 is a double eigenvalue" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "quo(p,(L+15)^2,L);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,&%\"LG\"\"\"\"#D!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 21 "and  25 is the other." }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "A:=evalm(Asave-25);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%!#N\"#5!#:7%F+!#?!#I7%!\"&!#
5!#D" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A:=mulrow(A,1,-1/35
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"\"#!\"
#\"\"(#\"\"$F-7%\"#5!#?!#I7%!\"&!#5!#D" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "for i from 2 to 3 do A:=addrow(A,1,i,-A[i,1]) end do;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"\"#!\"#
\"\"(#\"\"$F-7%\"\"!#!$?\"F-#!$S#F-7%!\"&!#5!#D" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"\"#!\"#\"\"(#\"\"$F-7%\"\"!#!
$?\"F-#!$S#F-7%F1#!#!)F-#!$g\"F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "A:=mulrow(A,2,-7/120);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"AG-%'matrixG6#7%7%\"\"\"#!\"#\"\"(#\"\"$F-7%\"\"!F*\"\"#7%F1
#!#!)F-#!$g\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A:=addro
w(A,2,1,2/7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7
%\"\"\"\"\"!F*7%F+F*\"\"#7%F+#!#!)\"\"(#!$g\"F1" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 22 "A:=addrow(A,2,3,80/7);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"\"\"\"!F*7%F+F*\"\"#7%F+F+F+
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "The corresponding eigenvector
 is:   [-1, -2, 1]" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "evalm(Asave &* [-1,-2,1]);   #checking" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%!#D!#]\"#D" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 19 "A:=evalm(Asave+15);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"&\"#5!#:7%F+\"#?!#I7%!\"&!#5
\"#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "A:=mulrow(A,1,1/5);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"\"\"\"#!
\"$7%\"#5\"#?!#I7%!\"&!#5\"#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 53 "for i from 2 to 3 do A:=addrow(A,1,i,-A[i,1]) end do;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"\"\"\"#!\"$7%\"\"!
F.F.7%!\"&!#5\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6
#7%7%\"\"\"\"\"#!\"$7%\"\"!F.F.F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
55 "The two eigenvectors are:    [-2, 1, 0]   and   [3,0,1]" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "evalm(Asave &* [-2,1,0]);   \+
evalm(Asave &* [3,0,1]);   #checking" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#-%'vectorG6#7%\"#I!#:\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vec
torG6#7%!#X\"\"!!#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Q14:" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Asave:=matrix(3,3,[0,7,0,0,0,0,0,0,
-2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&AsaveG-%'matrixG6#7%7%\"\"
!\"\"(F*7%F*F*F*7%F*F*!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Eig
envalues are clearly  0,  0,  and  -2, since the matrix is upper trian
gular." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A:=evalm(Asave+2)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"#\"\"(
\"\"!7%F,F*F,7%F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "A:
=mulrow(A,1,1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6
#7%7%\"\"\"#\"\"(\"\"#\"\"!7%F.F-F.7%F.F.F." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 19 "A:=mulrow(A,2,1/2);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"AG-%'matrixG6#7%7%\"\"\"#\"\"(\"\"#\"\"!7%F.F*F.7%F.F.F." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "A:=addrow(A,2,1,-7/2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"\"\"\"!F+7%F
+F*F+7%F+F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "The eigenvector \+
corresponding to -2 is  [0,0,1]" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 42 "evalm(Asave &* [0,0,1]);    #just checking
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"!F'!\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "A:=evalm(Asave);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"!\"\"(F*7%F*F*F*7%
F*F*!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "A:=mulrow(A,1,1
/7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"!\"
\"\"F*7%F*F*F*7%F*F*!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 
"A:=swaprow(A,2,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrix
G6#7%7%\"\"!\"\"\"F*7%F*F*!\"#7%F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "A:=mulrow(A,2,-1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"AG-%'matrixG6#7%7%\"\"!\"\"\"F*7%F*F*F+7%F*F*F*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 70 "There is only one eigenvector correspondi
ng to  L = 0, namely  [1,0,0]" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 36 "evalm(Asave &* [1,0,0]);   #checking" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"!F'F'" }}}}{MARK "80
 0 0" 61 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 
}