Math 3013
Homework Set 9
Problems from §5.1 (pgs. 300-301 of text): 2,4,6,8,10,12,17,19,23
Problems from §5.2 (pgs. 315-317 of text): 1,2,3,4,5,9,10,13
Problems from §7.1 (pgs. 394-395 of text): 7,8,9,10
1. (Problems 5.1.2, 5.1.4, 5.1.6, 5.1.8, 5.1.10, and 5.1.12 in text). Find the characteristic polynomial, the
real eigenvalues, and the corresponding eigenvectors for the following matrices.
(a) A=7 5
−10 −8
(b) A=−7−5
16 17
(c) A=1−2
1 2
(d) A=
−1 0 0
−4 2 −1
403
(e) A=
100
−8 4 −5
809
(f) A=
−4 0 0
−7 2 −1
703
2. (Problems 5.1.17 and 5.1.19 in text) Find the eigenvalues λiand the corresponding eigenvectors vifor
the following linear transformations.
(a) T([x, y]) = [2x−3y, −3x+ 2y]
(b) T([x1, x2, x3]) = [x1+x3, x2, x1+x3]
3. Mark each of the following statements True or False.
(a) Every square matrix has real eigenvalues.
(b) Every n×nmatrix has ndistinct (possible complex) eigenvalues.
(c) Every n×nmatrix has nnot necessarily distinct and possibly complex eigenvalues.
(d) There can be only one eigenvalue associated with an eigenvector of a linear transformation.
(e) There can be only one eigenvector associated with an eigenvalue of a linear transformation.
(f) If vis an eigenvector of a matrix A, then vis an eigenvector of A+cIfor all scalars c.
(g) If λis an eigenvalue of a matrix A, then λis an eigenvalue of A+cIfor all scalars c.
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