21-241 Matrix Algebra
Homework #8:
Exercises: Due Wednesday, November 9
Section 5.3: 1, 3, 15, 18, 24, 27, 31
Section 5.5: 5
Problem A: Find all real and complex eigenvalues of the 3 X 3
matrix A below, and for each eigenvalue, find a corresponding
eigenvector.
[1 1 -1]
A = [0 1 0]
[1 0 1]
Problem B: Find all real and complex roots of the polynomial
x^3 + x^2 + 17x - 87
Hints: First look for real integer roots by substituting values
such as 0, 1, -1, 2, -2, etc., into the polynomial, in hopes that you
will find a
"nice" root r. If you find one, then you know that the polynomial can
be factored as follows:
(x - r) (x^2 + ax + b)
By multiplying these factors, you can determine the appropriate values
for
a and b. Finally, you can find the roots of the quadratic factor,
using the
quadratic formula if necessary.
Problem C: Find all real and complex eigenvalues of the 4 X 4
matrix A below, and for each eigenvalue, find a corresponding
eigenvector.
[1 -1 1 -1]
A = [1 1 1 1]
[0 0 1 1]
[0 0 1 1]
Section 5.6: 3, 10, 12, 14, 15, 17ab