Question

9. [10 points (A) True or False. Circle your answer and justify it by showing your wot (a) T F: Let A be any square matrix, t (b) T F: If S is invertible, then ST is also invertible. hen AT A, AAT, and A+ AT are all symmetric. If a row exchange is required to reduce matrix A into upper triangular form U then A can not be factored as A-LU (d) T F Suppose A reduces to upper triangular U but U has a O in pivot position, then A has no LDU factorization.

Answer 1

(A) (a) True[Therefore (A^T A)^T = A^T (A^T)^T = A^T A, (A A^T)^T = (A^T)^T A^T = A A^T, (A + A^T)^T = (A + A^T)^T = A^T + (A^T)^T = A^T + A = A + A^T (b) True (Because s is invertible implies set (s) notequalto 0 implies set (S^T) = set (s) notequalto 0 implies S^T is invertible] (c) True [Because If A required a exchange to reduce into U_1 then A has zero pivot. Therefore PA = LU & A notequalto LU (d) False [Because For A = [0 0 1 2], A = U[0 0 1 2] has zero pivot, but A = LDU, L = [1 1 0 1], D = [1 0 0 1], U = [0 0 1 1]] (B) (a) Let A = [3 0 9 7] then A = [1 0 0 1] [3 0 0 7] [1 0 3 1] is the required decomposition. (b) If A & B are symmetric, the following are also symmetric implies A + B, A - B, A^2, ABA, ABABA (c) The inverse of P = [0 0 1 1 0 0 0 1 0] is P^-1 = [0 1 0 0 0 1 1 0 0]