40 pts.
Problem 1. Let A be the matrix
"
−2
−2
6
5
#
A=
.
Find the characteristic polynomial of A and the eigenvalues of A by hand
computation.
70 pts.
Problem 2. In each part, you are given a matrix A and the eigenvalues of
A. Find a basis for each of the eigenspaces. Determine if A is
diagonalizable and, if so, find an invertible matrix P and a diagonal matrix D
so that P −1 AP = D. [The use of a calculator is highly recommended. Say what
you put into the calculator and what you got out.]
A. The eigenvalues are 1 and 2 and the matrix is


2 −1 1



A=
 1 0 2 .
1 −1 3
B. The eigenvalues are 1 and 2 and the matrix is


2 −1 1


1 1 
A=
.
 0
0
0 2
70 pts.
Problem 3. Consider the system of differential equations
(3.1)
x01 (t) = x1 (t) + x2 (t)
x02 (t) = −2x1 (t) + 4x2 (t)
This system can be written in matrix form as x0 = Ax where A is the matrix
"
#
1 1
A=
.
−2 4
This matrix is diagonalizable, in fact P −1 AP = D where
"
#
"
#
1 1
2 0
P =
, D=
.
1 2
0 3
[The use of a calculator is recommended on this problem. Say what you put
into the calculator and what you got out.]
1
A. Use the information above to find the matrix etA .
B. Find the solution of the system (3.1) for arbitrary initial conditions
(
x1 (0) = x01
x2 (0) = x02
C. Find the solution of the system (3.1) for the initial conditions
(
x1 (0) = 2
x2 (0) = −1
2
EXAM
Exam 2
Math 3351, Fall 2010
Nov. 3, 2010
• Write all of your answers on separate sheets of paper.
You can keep the exam questions when you leave.
You may leave when finished.
• You must show enough work to justify your answers.
Unless otherwise instructed,
give exact answers, not
√
approximations (e.g., 2, not 1.414).
• This exam has 3 problems. There are 180 points
total.
Good luck!