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