13 11 points previous answers holtlinalg1 61046

13.
1/1 points | Previous Answers HoltLinAlg1 6.1.046.
Determine if the statement is true or false, and justify your answer.
The product of the eigenvalues (counting multiplicities) of A is equal to the constant term of the
characteristic polynomial of A .
Solution or Explanation
False. Consider .
10
01
det( A −
λ
I
n
) = (
λ
1
−
λ
)(
λ
2
−
λ
) (
λ
n
−
λ
),

14.
4/4 points | Previous Answers HoltLinAlg1 6.1.047.
Suppose that A is a square matrix with characteristic polynomial
(a) What are the dimensions of A ? (Give n such that the dimensions are
(
λ
− 3 )
2
(
λ
− 4 )
4
(
λ
+ 1).
7
(b) What are the eigenvalues of A ? (Enter your answers as a comma-separated list.) (c) Is A invertible?
(d) What is the largest possible dimension for an eigenspace of A ?
4 4
Solution or Explanation
(c) A is invertible, since 0 is not an eigenvalue.
(d) The largest possible dimension of an eigenspace is 4 , corresponding to
.
1.
.