QUESTIONS:
1)( 40 pts.)Let
be given.
(a)(5 pts.)Find all eigenvalues of A.
(b)(10 pts.)Find a basis for each eigenspace and determine the associated eigenvectors.
(c)(5 pts.)Find an invertible matrix P and a diagonal matrix D such that P−1AP = D.
(d)(10 pts.)Find the inverse of the matrix P in part (c) by using Gaussian elimination method. Verify your answer i.e. show that PP−1 = I = P−1P.
(e)(5 pts.)Calculate A2020.
(f)(5 pts.)Find A−1 by using Cayley-Hamilton Theorem.
- (a)(10 pts.) The characteristic polynomial of a certain 3x3 matrix A is p(x) = x3 − 7x2 + 5x − 9. Use this fact to express adj(A) as a linear combination of A2,A and I.
(b)(10 pts.) If A is an nxn non-singular matrix, show that adj(A) can be
expressed as a linear combination of An−1,An−2,…,A,I.
- (10 pts.)The nxn matrix A is said to be idempotent if A2 = A. If λ is an eigenvalue of such a matrix, show that λ is either 0 or 1. What can be said about a non-singular idempotent matrix?
