Problem 1. Let A be an invertible n X n matrix. If A is diagonalizable, prove that A1 is diagonalizable. Problem 2. Let A be a 2 X 2 matrix with exactly one real eigenvalue A, and assume A has geometric multiplicity 1 (note that this means A is not diagonalizable), (a) Prove that A AI 79 (8 8), but (A AI)2 = (8 8) (Hint: Use problem 44.25 from Homework 8.) (b) Let u E R2 be a nonzero vector which is not an eigenvector of A, and let 1) = (A AIM. Prove that v is an eigenvector of A, and that {u, 1)} is linearly independent, /\\ 0 (c) Prove that A is similar to the matrix (1 A ). (Hint: Consider the matrix S with columns u and 1). Problem 3. Let A be a 2 X 2 matrix with eigenvalues A1, A2 (not necessarily distinct, not necessarily real). Prove /\\1 + A2 = tr(A) and MM = det(A) (recall the trace tr(A) of a matrix is the sum of the diagonal entries)