{VERSION 3 0 "IBM INTEL NT" "3.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 255 0 123 
0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Ou
tput" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 
0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE 
"" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 
0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 
0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 
1 {CSTYLE "" -1 -1 "" 0 1 0 0 2 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 
0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 
0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 
{CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 
0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 
0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 
{CSTYLE "" -1 -1 "" 0 1 0 0 2 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 
0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 
0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 
{CSTYLE "" -1 -1 "" 0 1 10 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 
0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 
0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Matrix Algebra investigati
on 3" }}{PARA 0 "" 0 "" {TEXT -1 8 "Consider" }}{PARA 256 "" 0 "" 
{OLE 1 4097 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
fyyyyya:nYf::G:jy;::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n<Ejyyiy=C:Qj;>:
;JZC:bKB::>`Gvt]_yWpq;V:>j>Z:j:vCSmlJ::::::::::OJ;@jyyyyyy;jysy;Z:::::
::^<>:F::]KRnC=MtFGgml>:;::::::JGN:ry:>:<::::::=J:^F>:F:AlqfG[maNFO=;:
:::::::_J;vyyuy:>:<::::::AJ:<JhJ:JyK;B:J<:n:v:<J>>:OJ:vYxy?J@j@>:W:YJ:
>\\:B:]:wAyA::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::j:b:<Z::::::::::JjB:CR:Hj<dj:Dk;AN;TrXHj;R:ER:=R:?b:FZ<>
kmR:?jRFZ<njgNZBH@>K<J:DJEiO;TshJb<RMt><::::::::::=B:wyyAb<vYxy?V:<J:>
R<:TNC>Z:fc[_hb_ds?h_?^?GhoGfnGgioGM:ETV:::^xsNpk[::::::::jysyA:CB:F:;
jysy?:;:QF<F]?JGE:=j>r:UT:NZ:f<<J:f:;j>D:;B:C:?R:=j<j:Djyyyy=j<j>Z:^:n
_;>=E:]c:=Z:f:V[<Z:f:^[<>n>>rAn;n^@v;_jysy;B:KZ:>ryYnyAjAVb>V;n>^;UTR:
;Jrg:<j>J;<ja^G>D_mlVH[KRJ:<:::::::>=?R:>Z:N:;B:yayA:;B::::::f:<J[E:<:
?ja:[LsjsJRJ:<:::::::>=;jysy:>:<::::::wqy[:::::::::::::vYxI:;Z::::::::
::::::::::::yay=J:B::::::::jD<Z:Vy<>jx]:JBAj:J:DZJ^dcgg_WhZnc_whZNdigg
[oGuJ<jP@:MB:;B:_c<;h:Cm:>Z:>:::::::::::FB[QBB:qAB:>l;>:DZ<B>aTXDpql`P
^:f?=J<JrA:MJ:N`DJLAj:>Z:fAkp>JSfu=Ni=F:;jP<Jf?<<jw?Zw]:JBA:<Z<B:GTSWu
REuVTK<jP@J:^:>X=j>>:_KZAJiEj:>Z:^C[F:MZ:N@yC;wh:=:WLBG;<JSF`=nu<F:fc:
>LMZ:N@;plUj:>:YLB=j>B:_kYfv@F:FD[n>B:_kRfv@F:>D[nDB:qQ:[:JBA:<Z<B>c\\
_;NDjp^:f?=J<Jr?:MJ:<JSfm;^u=F:<jS>RM:_;wc;=:]C:;@N@uPcLj:>:];JSJYTj:j
RJ?B:G;OjyyiyI:>:[Z:VY[B:IB:>:sg:>Z:F:?bBaTXaEWEUUtP<jP>:C:[q:VZ:B:;:=
:::U;Jr;>:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::4:" }}{PARA 257 "" 0 "" {TEXT -1 8 "tr(A)= 4" }}{PARA 258 "" 
0 "" {TEXT -1 9 "det(A)= 4" }}{PARA 259 "" 0 "" {OLE 1 3585 1 "[xm]Br=
WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::wyyyqy;::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::NDYmq^H;C:ELq^H_mvJ::::::::gj<vyyuyJ<j?V:;J:>R<:T><::[S=ixRsYa
wW:A:;h:>Z:j:vCSmlJ::::::::::OJ;@jyyyyyy;jysy;Z:::::::^<>:F::]KRnC=MtF
Ggml>:;::::::JGN:ry:>:<::::::=J:^B>:F:AlqfG[maNFO=;::::::::_J;vyyuy:>:
<::::::AJ:<JhJ:JyK;:^::G:IZ:>;F;N;;Jyky;::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::=Z<B:<::::::::::;\\:^Z;
r:Eb<=b:EFKB:pnjLQnLov<qlvyy?;EKSJ::::::::::::::::=Jyyy;d:yayY;A:;JZ=J
:<:TNC>:U<RS;SK]UW=EWMuUWm>j\\@A:::cYH_WV>Z::::::::jysyA:CB:F:;jysy?:;
:IW:Ns;JI@:=j>r:Y>?JB>Z:j<>:Mb:B:C:?R:=j<j:Djyyyy=j<j>Z:^:n_;>=E:]c:=:
E:Qb:B:E:Sb:;W:;V:WB:n^@v;_jysy;B:K:;xy;xy;T:;U:Q:G;Sj`@@J:<JJ<j<B:M:?
B:YLpJbNHEms>@[C:>Z::::::::kJ;@:;J;>Z:vYxY:>Z::::::j<B:GT:<Z:J;vCJbNHV
H>@>Z::::::::kJ:vYxI:;Z::::::JywYB:::::::::::::yay=J:B::::::::::::::::
:::jysy:>:<::::::::>\\:B:qi:;fyB:>L;Z<b:<kc`qpDpql`]^:f_;j><J:<JSdJj@J
YF:<J\\>x:N;<JMJ?vyyuy=:;JBB:qQBv:;J:^q<Z:FZ<NZ\\VdsWhngfgcUC:Uk:^:>x;
j;<Z:>::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
2:" }{TEXT -1 3 "= 0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "To find t
he eigenvalues" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 
"" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 
"" {TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 26 "A:=matrix(2,2,[3,1,-1,1]);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 26 "eigenvalues:=eigenvals(A);" }}{PARA 11 "
" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "There is o
ne real positve repeated value." }}{PARA 0 "" 0 "" {TEXT -1 95 "If the
 matrix A is changed so that tr(A)= -4 what do you think would happen \+
to the eigenvalues?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 30 "Now consider a diagonal matrix" }}{PARA 260 "" 0 "" {OLE 
1 4097 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy::::::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::fyyyy
ya:nYf::G:jy;:::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n<Ejyyiy=C:Qj;>:;JZC:
bKB::>tM@u]_yWpq;V:>b>B:<:=ja^GE=;:::::::::N;?R:yyyyyyA:yayA:<::::::JD
J:j::F@[KaFFcmnnHEM:>:::::::oJ;Zy=J:B::::::F:;JrB:j:VBYmp>HYLkNG>:::::
:::N<?jyyiy=J:B::::::V:<JhJ:JyK;B:J<:n:v:<J>j>J?>:QJ:^;f;;JAjA>:[B:<Jy
ky;:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::=Z<B:<::::::::::SM:^Z;r:Eb<=b>AVJ?b[us:AZ;fZ;FZ;NZ<j:DJ=qL;\\RSF
Z<>kjR::G>>Z:NZPXe<nlW[HC:<::::::Jd\\jgB:;::j:<Jyyy;d:yayY;AZ:>:;`:Z@O
<;j`@Pt\\Pd`QrPPJPnrPqjLqnxPqF;fbk;::JtaMSAA;B::::::::vYxy;J<<:=J:vYxY
;J:Jpa:mR;^o<j:F;HJe@Z:NZ:f<<J:f:;j>D:;B:C:?R:=j<j:Djyyyy=j<j>Z:^:n_;>
=E:]c:=Z:f:V[<Z:f:^[<>n>R<W:Gc;YJCvYxY:<J>JZyyKyYZ]j[Lj?JMJ@fc[;>:[q<B
:M:?B:YLpfF[LsfFaMR>@>Z::::::::kJ;@:;J;>Z:vYxY:>Z::::::j<B:ch:B:N:Y<>D
_=a=[;;B:::::::JF>:yay=J:B::::::nYyA<::::::::::::jysy:>:<:::::::::::::
::::::vYxI:;Z:::::::::eB:<jwEJ:uI<:[V:=:;b:<Kd`ppPps<LaPpu<Lcxpp@Pq^GC
J:f_;j><J:<JSdJZEJlF:;B:=Lr?<<jw;<:[V:;Z<b:<kc`a\\wgfS?CB:f?=J<JrA:MJ:
N`DJLAj:>Z:fAkp>JSfm;Ni=F:;jP<Jf;_khIJ;Uj:jPJBB:qQ:pI<:[V:B:DZ:nb_og]g
fmS?C:US:;J<JrG:MJ:N@;X:wh:=J:<J`>l:F;<JSva>nu<F:nC[n>B:_kRHJiEj:j`<JB
G;<JS>REC<=:YLB=j>B:_kYf^BF:FD[n>B:_kRf^BF:>D[nDB:qQ:[:JBA:<Z<B>cTTPps
:_<F>C:Uk:^:>X;j>>Z:N@uNcLj:>Z:B:;p>ZDJrUj:JR>r::_kh?w=F:>@:_;ot;=B:B:
N;<JMJ?vyyuy=:;JBB:qQBv:>:sg:>Z:F:?bBaTXaEWEUUtP<jP>:C:[q:VZ:B:;::::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::6:" }}{PARA 261 "" 0 "" {TEXT -1 9 "tr(A)= 4," }}{PARA 262 "" 0 
"" {TEXT -1 9 "det(A)= 4" }}{PARA 263 "" 0 "" {OLE 1 3585 1 "[xm]Br=Wf
oRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::wyyyqy;::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::NDYmq^H;C:ELq^H_mvJ::::::::gj<vyyuyJ<j?V:;J:>R<:T><::[x^kxRsYawW
:A:;h:>Z:j:vCSmlJ::::::::::OJ;@jyyyyyy;jysy;Z:::::::^<>:F::]KRnC=MtFGg
ml>:;::::::JGN:ry:>:<::::::=J:^B>:F:AlqfG[maNFO=;::::::::_J;vyyuy:>:<:
:::::AZ:^E>:nYN::C:J=j=B:K:M:OJ:nYvY::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::j:b:<Z::::::::::JZB:CR:Hj<d
j:Dj<M><ZGGfn?gNGijgFyyYKjLN@>:::::::::::::::j:nyyYZDjysyQj;J:>r:>Z:Z@
O<;j`:PPJPnrPqjLqnxPqF;fbk;::JtaMSAA;B::::::::vYxy;J<<:=J:vYxY;J:jMA:O
X:n];j:F;HjAK;><;B:f:;j>D:<J<J;@j:f:FZ<vyyyy:f:F;B:C:WS:kj<jRDj:j<j?D:
<j<J@DJJAJ:AJA<:Gc;YJCvYxY:<J>JZyYZyYZ@Jj@j?JMJ@fc[;>Z:>^:f:<j>J;<ja^G
>D_mlVH[KR<:;B:::::::JFNZ;J:N:;B:yayA:;B::::::f:<J]@Z:B:N:Y<>D_=a=[;;B
:::::::JF>:yay=J:B::::::nYyA<::::::::::::jysy:>:<:::::::::::::::::::vY
xI:;Z:::::::::[B:<jwEJ:uI<:[N:b:DZJVdsgg\\wgfsBC:US:F[:>Z:N`D>f;nA=Z:^
B[I:OZ:n>N;yyyxy:J:><<jw?<IJ:>:sg:B:=b:?bBaTXaEWEUUtP<jPF:C:[Y:VZ:B:;:
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::6:" }
{TEXT -1 2 "=0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "B:=matrix
(2,2,[-2,0,0,-2]);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 26 "eigenvalues:=eigenvals(B);" }}{PARA 11 "
" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "There is o
ne real negative repeated value." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
105 "If each -2 is changed to 2 tr(A) will become positive. What do yo
u think will happen to the eigenvalues? " }}{PARA 0 "" 0 "" {TEXT -1 
139 "Note that even though there are an infinite number of eigenvector
s associated with a diagonal matrix there is only one repeated eigenva
lue." }}{PARA 0 "" 0 "" {TEXT -1 56 "You should now be able to complet
e the eigenvalues table" }}}}{MARK "14" 0 }{VIEWOPTS 1 1 0 1 1 1803 }