1. Find the eigenvalues and corresponding eigenspaces for each of the following matrices: 2. Let A be an n x n matrix (a) Prove that A is singular if and only if A = 0 is an eigenvalue of A. (b) Lal A be a nonsingular matrix and let. A be an eigenvalue of A. Show that 1/A is an eigenvalue of A-1. How do we know that A * (?. 3. An n x n matrix is said to be idempotent if A? = A. Show that if A is an eigenvalue of an idempotent matrix, then A must be either 0 or 1. 4. In each of the following, factor the matrix A into a product XDX ', where D is diagonal: 0 ( 6) 2 1 1 5. For each of the matrices in Problem 4, use the X DX-1 factorization to compute A-1. 6. Let A be diagonalizable matrix whose eigenvalues are all either 1 or -1. Show that A-1 = A. (Hint: Factor A into the form XDX-'.)