Let A and B be square matrices of the same size. Prove that if A and B are both upper triangular, then AB is upper triangular. If A and B are both upper triangular, do they necessarily commute? If you think they must commute, give a proof; if you think they might not commute, give a counterexample.

PODASIP the following statements about products and inverses of triangular and unit triangular matrices. a. The inverse of a triangular matrix is triangular. b. The product of two triangular matrices is triangular. c. The inverse of a unit upper (lower) triangular matrix is upper (lower) triangular. d. The product of two-unit triangular matrices is unit triangular.

Show that is A is a square matrix, then A + AT is a symmetric matrix. You can use any matrix for A as long as it is a square matrix.

I believe A is correct, but I am not sure about w. It needs to be written as a fraction. Thanks!