PROBLEM SET
Problems on Matrix Exponentials
Math 3351, Spring 2011
March 29, 2011
• Write all of your answers on separate sheets of paper.
You can keep the question sheet.
• You must show enough work to justify your answers.
Unless otherwise instructed,
give exact answers, not
√
approximations (e.g., 2, not 1.414).
• This problem et has 3 problems. There are 0 points
total.
Good luck!
These problems assume the use of a TI-89 (or similar) calculator. To find the
eigenvalues of a matrix, use the calculator to find the characteristic polynomial
and use the solve or csolve functions to find the roots. Use the calculator to
find RREFs and powers of matrices.
The eigenvalue and eigenvectors functions on the calculator probably won’t
work on these problems because they only give approximate answers.
Express your answers as fractions, not decimals.
In the answers, exp(t) is often used in place of et
Problem 1.
In each part, you are given a matrix A which is diagonalizable. Find the
diagonalization and use it to find etA . If an initial value problem is given, find
the solution.
A.
2
A = 0
0
B.
A=
0
5
0
6
−3
0
0 .
1
4
−1
.
Solve the initial value problem
x0 (t) = Ax(t)
−2
x(0) = c =
.
3
C.
0 −492
−1 −264 .
0 −121
122
A = 66
30
D.
A=
2
3
−3
2
.
Solve the initial value problem
x(t) = Ax(t)
−3
x(0) = c =
.
2
Problem 2.
In each part, you are given a nilpotent matrix N . Verify that N is nilpotent
and compute etN .
1
A.
1
0
0
537
N = 215
94
257
103
45
0
N = 0
0
B.
C.
0
1 .
0
−3656
−1464
−640
336
21
84
84
−1584 −20
928 −416
.
N =
79
0 −63
21
−1027 −79 −505 −253
Problem 3.
In each part, you are given a matrix A which is (definitely) nondiagonalizable. Show that the matrix is nondiagonalizable and find the Jordan Decomposition A = S + N . Use the Jordan Decomposition to compute etA . If an initial
value problem is given, solve the initial value problem.
A.
323 −468 1140
88
A = 25 −37
−81
117 −286
Solve the initial value problem
x0 (t) = Ax(t)
−1
x(0) = c = 1
2
B.
25
12
A=
−2
82
C.
1
2
A=
0
0
38 2
20 1
−3 2
130 7
−12
−6
1
−40
−2 1
0
1 0
1
0 1 −2
0 2
1
2
0
