1) Let
be given.
(a) Find all eigenvalues of A.
(b) Find a basis for each eigenspace and determine the associated eigenvectors.
(c) 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) Calculate A2020.
(f) Find A−1 by using Cayley-Hamilton Theorem.
2) (a) 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 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.
3) 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?
