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