Homework 9
Due: Wednesday, November 5
1. Consider the matrix A =
4 −1
6 −1
(a) Compute the characteristic polynomial p A ( x).
(b) Determine all eigenvalue(s) of A.
(c) For each eigenvalue, compute the corresponding eigenspace ε A (λ).
(d) What are the algebraic multiplicities of the eigenvalues?
(e) What are the geometric multiplicities of the eigenvalues?
(f) Based on the previous two parts, is A diagonalizable?
(g) Based on (b), is A invertible?
1 2
2. Repeat Problem 1 with the matrix B =
.
2 4
1
1
3. Repeat Problem 1 with the matrix C =
.
−1 −1
4. Let λ be an eigenvalue of matrix A ∈ Mat n,n (F).
(a) Suppose α ∈ F. Show that αλ is an eigenvalue of α A. (H INT: Use the definition of
eigenvalue.)
(b) Show that λk is an eigenvalue of Ak . (H INT: Ditto.)
(c) Suppose p( z) is a polynomial with coefficients in F. Using the previous parts, show
that p(λ) is an eigenvalue of p( A).
5. Prove that the constant term of the characteristic polynomial p A ( x) of a square matrix A is
det( A). (H INT: Let p A ( x) = det( A − xI ) = a0 + a1 x + · · · + an xn and consider what happens if
x = 0.)
Professor Dan Bates
Colorado State University
M369 Linear Algebra
Fall 2008