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

[Reserved for Score]

— 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

[Reserved for Score]

[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