C 3 points a has characteristic polynomial f a ? 3 ?

This preview shows page 8 - 9 out of 9 pages.

(c) (3 points) A has characteristic polynomial f A ( λ ) = (3 - λ ) λ 2 .
(d) (3 points) A has characteristic polynomial f A ( λ ) = (3 - λ ) λ 2 and rank( A ) = 1.
Page 7 of 8
Math 217 Final Exam December 12, 2014 8. (a) (5 points) Let A be a diagonalizable n × n matrix. Prove that for any k R , the matrix A + kI n is diagonalizable.
(b) (5 points) Let A be a skew-symmetric n × n -matrix with characteristic polynomial f A ( λ ). Prove that f A ( - λ ) = f A ( λ ) if n is even, and that f A ( - λ ) = - f A ( λ ) if n is odd.
(c) (5 points) Let A R n × n and define T : R n × n R n × n as the linear transformation given by T ( M ) = AM for all M R n × n . Prove that T and A have the same eigenvalues.
Page 8 of 8

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture