Linear algebra question-matrices and symmetry

Question

Question:

Linear Algebra Question: Matrices and Symmetry

Assume that A and B are two symmetric k x k matrices.

a) Show that

                                    AB = (BA)^T

b) Show that if A is invertible, then A^-1 is also a symmetric matrix.

c) Assume that we can write
                                        A= [ A₁  A₂ ]
                                             [ A₁  A₂]

With A₁ a 2 x 2, A₂ an m x 3, A₃ a 3 x n and A₄ a p x q matrix. What can we say about m, n, p, q? Which of the four submatrices is symmetric? Can you say anything about the relationship between A₂ and A₃?