F16 10 :10 Exam 1 Prob lem 1Linear Algebra, Dave Bayer
Test 1
N a m e .
IIIII IIItes t laZp l
U n i .
[1] Find the determinant of the matrix
de(r(:M=
= ' 3 .
I S MI I I IM 3 II I I ^
00 O 0® t 11
o o o®
" 1 1 5 1 1 "
'G 1 1 1 1 1— A = 0 3 0 0 0
f - - h 1 1 1 3 1— 1 1 1 1 4
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— 4 "
- 1 2 - ^ - 7 2
F16 10 :10 Exam 1 P rob lem 2Linear Algebra, Dave Bayer
Tes t 1
I III III IItestla2p2
[Reserved for Score]
[2] Find the inverse to the matrix
A =
2 3 4 '3 2 11 1 0
2 3 1 2 - 33 ^ 1 3 2 -4 1 0 H 1■2.
3 - 2 . I 3 2
r - i H -5 'l - M \ 0I I - 5
/5
F16 10 :10 Exam 1 P rob lem 3
Linear Algebra, Dave Bayer
Test 1testlb2p3
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[3] Using Cramer's rule, solve for y in the system of equations
' a 1 2 " X " 1 "b 1 1 y = 1
rHO1 z 5
q I ^h 1 Ic 5 4
a I tb I 1c I q
CA 5412^1\
a iql-^ \ zm
I z -11
— G b - c
5 ( x - Z b - C
(h \ j) ̂ + (lE ).'̂ (ED)c' (rs)^.(e2)^^(eE)c
' % \ Z '" 1 1 ^'1
2. I I H U 1
G I q. 5^ " J
c-6 3
- Z + I 2 - GG ' H - Q
= -1 d
CUoQ̂ ̂/b,c uvieve wt KMouithesolo)
F 1 6 1 0 : 1 0 E x a m 1 P r o b l e m 4
Linear Algebra, Dave Bayer I [Reserved for Score]
testla2p4T e s t l
[4] Find a system of eigenvalues and eigenvectors for the matrix A, and find a formula for A"^, where
A =
Xi>2.= ^'9-2-1-18
4 1
2 5
Q-
2 . 2 - I
- 2 I ^
Check ' ^ 1r r'7^ I
of
Y S - \
A
^ /3' 3 .
F16 10:10 Exam 1 Problem 5Linear Algebra, Dave Bayer
T e s t l
III miltestla2p5
[Reserved for Score]
[5] Let f (n) be the determinant of the n x n matrix in the sequence
[ ]4 1
3 4
4 1 0
3 4 1
0 3 4
4 1 0 0
3 4 1 0
0 3 4 1
0 0 3 4
" 4 1 0 0 0 "3 4 1 0 0
0 3 4 1 0
0 0 3 4 1
0 0 0 3 4
Find a recurrence relation for f (n). Express f(n) using a matrix power. Find a formula for f(n).
A - I -
A - 3 r -
■-I I
I ^-?> I
^ (i5j) f-1) + (H3j) - 2)f ( n ) [OH M
n
" f (0)"_ f (n+ l )_ _f(i)_
f{n) = 4 ■*" n