Final exam solution key
M E T U f)epartment of Mathematics
MATH
Date
Time
:
260 FINAL
EXAM
Last Name
May 26, 2015
Name
Student No
Signature
: 17:00
Duration : 110 mzn.
4 Questions on 4 Pages. totally'80 points
Keep your ID out during the exam
SHOW YOUR WORK
Q.1. (16 pts.) a) (S pts.) Let A. B, C,D, and Ebe
3
x 2 matrices such
t*
C
and E -2R2:Rr n
\\irite dou'n an invertible matrix P such that PA : -e as a product of four elementar-v
A
R:!$'
3 n':4R' C
matrices (accordingl)' to the diagrams above).
h
Pb ?j'c ?DT*
P:
97.3
E
l?z
lo*
€r() €,ft)€,&) €,t)= Lou
il Li:flE?AEn
(n > Li-3?71
t t oJ
b) (4 pts.)
a sl
la
r
t)
Given't'
^"1sl
lo a 2l9
t i q*3
I+
=Y Iz
0
!
q
9r
Tx)
Z {,'
a
-d
)
Compute the de termrnant
a*3 b 3r3l
-t
7r' rt
5.ro
l.
2 a 2ttl
Etq 911
t
rl
i(
r'/JiHq zl
8 zl
"
r
lzq37P3x.
(4 pts.) Let )3 +2^2 - ) - 5 be the characteristic poll-nomial of a matrix ,4. \\rrite the
")
matrix ,4-1 as a linear combination of the ma,trices 1, A. and A2. Is the set {A, At , At , An}
frL$r?u Jr
5+-,
ptr.) Determine u'hether or not tire follou'ing matrix is diagonalizable. and if
is so, fi.nd a diagonalizing matrix P and a diagonal matrix D such that P-r AP : D.
Justif;,- your ans'frrers.
Q.2. (24
it
Io0
3 -8 I
a) (12 pts.) .4: | -2 0 |
Li 3 -3
.
1
=
()*C 63e ). *
gu, cu[j
+
4r.1
n)
(1r, A3) 61p ), ) .b3 li-F
=
5 l,*^
'rfh*ffi
o=-7"!Jrr&xffia
Vqr,q2&
ervwlal^.,o'^,s
of egxu;trec+-'Ds oI d;u*\it"
l&."& A .l;+ olrqBoua('ga-Ws
i;'*rd
b)
(12 pts.
u: l-z
' L1
I )+z o
llLBl= | -t
-":l
o
-I
_lI
;
)-51
&--^:s -roo o
I
t-
I
t-ir-el = L: ^o:ll
L
4" \=5
a
6*z)(x* z)[x -s)
I ),=l!-, , )r: r
?o=H ,?"-F]
->
+8;
Ln
il
9 ,,Z'nnQ
4w
:
I sJ-BJ = L:l
-
4- <
o ol
>+z-T
f-r 0
otlts
l-lt;iil'wT
f-?
Z 3J
B
q=I?]
e B. (zflpts.)
Let
_:
A:
[
S I]
a) (16 pts.) Find a diagonal matrix D and an orthogonal matrix
lxtrAl=
[i
j?_,\ :6-r)./r*,):(r-,)(r*,)=
eiyvva&*s: )r={-
$?,=\r
Q such Lhat Qr
) tr"=\g=-1
AQ
:
D.
i.
6=)
-
=rfl
q'=[]
,
;l - fr\il,E=El
F,i-
{^' )=\1},
fj
F.ro]=
srr,,cR
f r rs ...t b
(
j :l-W:il D r,=El ,tF\
",iq; ^r};fr:Yffi^utr)
Q.= LH,
Q,=lil
q =)!,u,nJ
u) (5ntg).
rila
n*h
t
-
{tt\il
a diagonal matrix D1 sucir that
Qr(.43
matrlx u'hich is found in a).
ahi>l)Q=
(
+
)Q: Ir
3A2
in'here
Q is ttre
rT ..Z
')
q?qr2q;h=&)l q)* 3(q4E-
= f- >;= H',:p,,fl,J
o01
zo
fto z)
I
Q.4. (18pts.) LetPs bethespaceof all polr'nomialsof degree(3,andietI:P3-+Ps
be a linear transformation such that I(1 + *) : 2, L(l - 3t ) - -6, L(t + 2*) :4. and
L(t3+2t):61.
U9=hftrt')- r) = l(n+)-l1r)= z-tc : 2
L(l) =/u (Gru"i -zt):rbra)-zl(1)= +'?z= o
L
b) (3 pts.)
Write a basis for
J* (t l =="';;" il
Jnrlr) : {r(p) i p e Pr} ^,
ri,;lr) r' l'i;;,'
qo a^r- *o/lor",q. {2, (4e\
.)
(3
pt".)
Find the dimension
;:ifi =
,^
eryu
q (eere -t")
of the linear transformation
u-ith respect to the standard ordered basis of 23.
S,,,cQ Kbh (lr\: T*r tr.)
4
o
0
o
{7 # I
{-Ilj=q
of Ker(l) : {p e Pzl L(p) - 0}. Justify your answer.
d) (6 pts.) Find the matrix representation
Lrl = frr{=
^n
g
pcq,
o-qo1
L
o-l2l
o 5 0l
0 o A)
It[ : I -2L*3L2
Jt*r', ),'= Q G eurl- -l
prl{l, *D qrFt+olJ=
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