4. Let A be a square matrix. (a) 1² is an eigenvalue of A² with the same eigenvector, v. Show that if A is an eigenvalue of A with corresponding eigenvector v, then (b) What is the characteristic polynomial of A²? Suppose that the characteristic polynomial of A is fa(^) = (A + 2)(A – 3). %3D Suppose that v is an eigenvector of A². Is it necessarily true that v is also (c) an eigenvector of A? If so, then prove this fact; if not, then give an example that shows this is false.

in 2) Suppose that A is a square matrix and let p(x) be a polynomial with real coefficients with variable x. If p(x) = E a;x' i = 1 in , then define p (A) = E a¡A' . If, is an eigenvalue of A, the show that p (1) is an eigenvalue of p(A). i = 1

Let A = = -8 2 2 2 2 -5 4 4 (1) Compute the characteristic polynomial of the matrix A. (2) Find the eigenvalue of A and their multiplicities. (3) Is A invertible? Why?

1 Let P = -2 - Find p P. Deduce P, if it exists, from the result of -1 p' P. You can also calculate P 1 using the traditional approach but this will be slower. 2. Let A be a 3 x 3 matrix with the following eigen-pairs: 1 (1, ):(1, 0 ):G -2 ).