a Give an example to show that if A and B are symmetric n x n matrices then AB need not be symmetric 02 OA 2 3 B 1 4 which is not symmetric OA 3 2 t B 10 23 OA 20 13 20 OA A OA B B but AB but AB but AB but AB 02 2 81 0 4 81 0 01 1 but AB which is not symmetric which is not symmetric which is not symmetric which is not symmetric b Prove that If A and B are symmetric n x n matrices then AB is symmetric if and only if AB BA O Suppose A and B are symmetric Then AB AB and BA BA Thus AB is symmetric If and only if AB AB BA E Suppose A and B are symmetric Then AB BTAT BA Thus AB is symmetric if and only if AB AB BA Suppose A and B are symmetric Then AB ATBT AB Thus AB Is symmetric If and only if BA AB AB Suppose A and B are symmetric Then AB BAT BA Thus AB is symmetric if and only if AB AB BA Suppose A and B are symmetric Then AB ATBT AB Thus AB Is symmetric if and only if 8A AB AB