1. No category

M E T U Department of Mathematics
Math 260 Basic Linear Algebra
Sprine 2O]5 Exam
Date
Time
:Apri.I62015
Last Name
: 17:10
Name
Duration
: 90 m'i'nutes
Department
:
:
:
Signature :
I
KEY
$tudent No.
Section
5 QUESTTONS ON 4 PAGES
TOTAL 60 POINTS
SHOW YOUR WORK
IN EVERY PROBLEM
:4
x-2A*z-t
2s-3a*22-3t :-1
a
3r-59*32-4t
z-L)l
:5
&tv
-aLa'-
Question 1 (10pts) Consider the system
a) Find the fundamental solutions
of the corresponding homogeneous system.
/ | -2
I n
-L'
I
in
'Z i 3 -i,$li^
-'2-
i
/
\--
,t
-
->
>
^7
')
!
n
:-
,l=t
/t.\
i:,
\2r
1
' (,/
l)
b) Write down the general soiution of the
4
_.)
2.
-!
it
-L
,a
-C
,
-a
.n, r
-+-
!(,1
+
i-t
z-
/
.li
*-
f--. 4
l
I
r-'.
t
^
/
..\
I
+?.1-
-J
I
j(-l'
'r- i,
?t,n/
J,
)
t
z,
a€ f.cc
,
ii
vrcr'(9V,j
J
i ;
-.otr-'.'
/: ir uX
ll
7h
'\.
1 > -7
t_v
-1_tJ
l-\-', 11
-Ji. ' ,tit,.;
:)
q
,1
iA
ttl
l.L
i \/
lt,'+li+
2l
tt
-Ll1
*
,
'
- l<.nL
'^
c) /
-
c 1i:f,
C--.f ct
J
(')z+(i
LetK:l 2., -r3
L-I 4 l.)
\L/
),11.
5ir!,
in terms of the fundarnental solutions found in part (a).
4./ -L
:
+ tL.
t; '' : .-'i
tl .ta
Tl* y*"{ i;1,':o. i
Question2 (bpts)
sy stem
*.-------)
nt'i
''rrt4r,
(? ),'\a {, i,, ,,,du:
(1
-! i L \ -t1.t/
.
''.;
-
11
r, t:_L/
,i-l : -qJ' Ii llnt
t c, ii-+
-t t -./ lA'.
'' i/'17 ,' - l'r''" ,
3
,-l?
- { :'4,,
l
i
)' {r)
wn"thecofactorsof Kandtheadjointmatrix adi(K).
r\
Li
l- r 1 1 ol
Question3(15pts)l,et,a:
3 S l.
| -r 0
[r I 1 r.]is row equivalent to ,4 and an invertible matrix P such that
a) Find a row reduced echelon matrix fi' which
R: PA.
} ITJ
8=#
jllj=lt
4 t ol ( c o\' (3Qt l,t L L ol t ,t D\
i-' e 3 5 o { t^, l-----+ /; L+5l4
I\J
V t 1_ I O (i l-t?iil,
c o ''\-l! Ut/
L 1_ o iL0 0 \ -i/.t k, I 5 --7 {: -5 -*l 4_/l '\
l-t_
t,
( t-;l
! + o Ie '( -)-t\ /
I J (
-i\,q
----) In
',,J Dr
\c c o Z l-t o L /
! -{ .-' r /
I
^tl\l
I
I
I
nt1
I
'{
i,-)
tz//
I
p_,
-..r
I
b)
Determine the va^rues of o and b such that the sysrem
t;
^lil:
I is consistent.
L;]1,]
12 r^u/ ror/;'rrdi $r,, c,l: {ht *ef {,r, rn{
h4 f c
Z'a/6
n' Ull
*[
,j
*/
,
c) Sotve the system
il
i-/n
i -'
\1
i',
/i
.ra..
,,-i
t
fawS
..(i.,.o/o,-c
'-., ,-(,,. t
J
€i<
l: IL ;
!_ I
o
,1
(,
lJ
)
L !
4
{
/^)
I
J
j
; --__-_---__\
+-,/ ='i ,l
Tt"
\,F) c e
{>
IJ
)-
-/
sho''al {r{r'i,;"' 4
L'",
/\
.'\
rl
\-!7
/
x-:z:7
! +*z = -3
i(o-3c
'nl/,J
',.
(_
/v
t
tui
IttlJiz- l,-
-/
-7
t/
,
4
-A.
cf,^r_'... f'
\_/
i'l rrr
olt,
LtI
)
,,n6{,,.*
I
a)
/
-f YI
-1-z-- -?)
//
!
.3\ t+
{-+\= +(4
t i- ' \''1o /I
\\.art
\
)
Question a (L5pts) a) Find the inverse
',';
,jlL 11
nl4
1I -------=s
4
nt
t,
Vt?4^t'iO ?
\.,/-
ltJ- *Vt
t.a
in
J"
.,
)
4(:
L/
| }-
-t
/-)f
!
il
\a
-)
1l
without using the determinant of B.
-1
3
t
o
_j
o {10
,fi
O
g----i s; _i_l 7 o -L/
,t n t-p, I Le' h{
"oi -/s
4.
\)L
T
,,t^
C
|. 3
Ir
OOI Li)N.rl2,l I
\JI
I
-q')t 1-I.r
.f E :
()t
L
/)\
,/l
/'1
c
z-- /
/
-V
i
-
i
--)4r't
3'e
tJ-
i
\
/t
U
4_t
^Lo
/)
/__
-,b i3 "z:
\-{
o Z/
(
lz o 1l["] :l?l-il
b)Giventr,uryrtu* l6 ; --t I I ; I
l,noasbyapplyingtheCramer'sruie'
o 1lL,J
l-r
LtJ
rule.)
ionswlthouttheuramersrule.,l
soiutions without the eramer's
i
l,nl a
trl
=
',
I
,t
t
I -r
o 7 o 5 sl
c) l,"t C: | ? 2 28 1 1 +l
lo o o o l rl
l,-l-:
r
-f t3
o0la
r.L4
r/
St
tvl
r II
w
I
!_
\J
r
r-n't,
, t-7 rt2oi
(
= 2('\'/l*rirl-:
t) ,t.l -,-,"r1 -
't:ii
(c /
t./-F
)-
,.'
-:
\;
li-
lt
)_!-
i
!(
t!
l*1
r-./
tntn
tz_{\J
t--t
.t
,,
-t".
--4
l!
I
l-.
LJ
I
t
-L=
i
4
''a
lul
1
i-\
I
I
11
t-
.,1
ial,g r
I
rlJ
{]-
/{
i,,A
!.
\>
l
Calculate det(C).
lo o o o 3 s.l
I ll=:
lc! = ltr{lHl tH!= l)>
_
a
jJ
I
/:t
\-/
(Nocred'itsfor
,,=\i
t'l I
o 3 tl
Is 6r 2r
21 1 1 2l
t:'i-
).
l-
;? :!lI -,.?
'-'
,/
I
_4
J
--4
-= 1 : -L
P
I
I
I
euestion b (lbpis) a)
Shorv rhar an lorver triangular
niatrlx is inrertibie if and only if ail eniries on its
diagonal are nonzero.
Aa+ h k uv,,l --lt tauSu!<t ,t',.q+ 2rx
Qr"+0+>
A Lb rh/rrr+ rkn <+ i Il= fi,,O zL . J,ogcutl eu*trc.g Q*o* o
e>
.
"Il
b) Given a matrix A, show that Ar A is symmetric-
i 4=A)r= r nf ( A')T=
{
l.r'=
{n
c) Given a matrix A, show that trace(ArA) > 0'
l:ic,(e- (nrn)
r-\I4e'\^yz
a;;;),=
f, i,o*A.r=
Z 1.,A.,=27 ta zo
=Z
J
K.-l
J=t
d)
:f
Assume that
A-'=
<)
E=l
A is inveriible. Prove that adl@) is invertible.
a4(4)
tr
thl
lU^
):l
a"\(*)
,
A: T
m i'AI = +,
)!+l ^'l:(l)
/+-*r.
5-.-t
lAl
'
,j
ou,l -'J*';*f =
-L
lAl
^4e\ ^oU
e) Assume that .4 is aa invet'cible n x n matrix. Calcuiate ladj(A)l in terrns of
ir{1.
A) =) lt-' l= lfotiA)l =
-r
h.
__Lr o\(tA)l g-+ a(sa 1=lrl =lA A
A-\4=
ndS(,
tR\n
t
-r'flAl'/ il
t^l
-r
lA
I'
/
6L /t'l : n,
I,Al)(A)l
I
{lzuu-
a*/ 1,4(t)l= lAl\-1