Question

Suppose that a is a square matrix with characteristic polynomial (λ − 3)2(λ − 6)3(λ + 1). (a) what are the dimensions of a? (give n such that the dimensions are n × n.) n = 6 correct: your answer is correct. (b) what are the eigenvalues of a? (enter your answers as a comma-separated list.) λ = 3,6,−1 correct: your answer is correct. (c) is a invertible?

Answer 1

The problem statement gives the correct answers for parts (a) and (b). The total number of roots of the characteristic polynomial is the dimension of the matrix: 6. The eigenvalues are the zeros of the characteristic polynomial, 3 (multiplicity 2), 6 (multiplicity 3), and -1.

(c) The matrix is not invertible when one or more eigenvalues is zero. None of yours are zero, so the matrix is invertible.