duction to Linear Algebra Examples II Problem 1. If p is a nonnegative integer and is a nonzero scalar, use mathematical induction to show that (A)p = pAp, for every square matrix A. Problem 2. For a nonsingular matrix A and nonzero scalar , show that (A)1 = 1 A1. Problem 3. Prove or disprove: For any m x n matrix A, AAT and AT A are symmetric. Problem 4. Show that if A is any n x n matrix, then the following hold. (i) A AT is symmetric. (ii) A AT is skew symmetric. Problem 5. Show that if A is an n x n matrix, then A = S K, where S is symmetric and K is skew symmetric. Furthermore, prove that this decomposition is unique. Z Make use of Problem 4. Problem 6. Show that if AB = AC and A is nonsingular, then the cancellation law holds; that is, B = C. Problem 7. Prove that if A is symmetric and nonsingular, then A1 is symmetric. Problem 8. Show that if the linear system Ax = b has more than one solution, then it must have infinitely many solutions. Z If x1 and