# MATH 410 University of California Davis Linear Algebra Exercises

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Math 410, Linear Algebra
Dr. B Truong
Name: _______________________
QUIZ CHAPTER 5
i) This quiz is worth 55 point. Please turn in in 5 days (Saturday 5/1)
− 9 − 6 − 22
1) (15) Let A =  1
2
2 

 4
2
10 
a) Find the characteristic equation of A
b) Find the eigenvalues of A
c) Find bases for the eigenspaces of A
2) (15) Find a non-singular matrix P such that A = PAP −1 is diagonal. Verify that P −1 AP is a
diagonal matrix with the eigenvalues on the diagonal.
 4 0 1
A = − 2 1 0
− 2 0 1
Use this result to find the value of A 4 . (1/5 credit for direct calculation!!)
3) (10) Solve the system of first order differential equations
− 4 2 
y’ = 
y
 0 − 2
1 1
4) (10) Which matrix is similar to 
 ?
1 1
A)
5)
 2 2
1 1 

 0 − 3
B) 

− 3 1 
1 − 3
C) 

0 1 
− 1 − 3
D) 
3 
1
(5) Prove that if y1 = e 1t x1 and y 2 = e 2t x2 are solutions to the system y ‘ = Ay , then
C1 y1 + C2 y2 is also a solution for real scalars C1 and C2
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Tags:
linear algebra

differential equations

characteristic equation

diagonal matrix

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