A matrix is symmetric if for each pair of indices i and j, the i, j entry equals the j, i entry. A matrix is antisymmetric if each i, j entry is the negative of the j, I entry.
(a) Give a symmetric 2 × 2 matrix and an antisymmetric 2 × 2 matrix. (Remark. For the second one, be careful about the entries on the diagonal.)
(b) What is the relationship between a square symmetric matrix and its transpose? Between a square antisymmetric matrix and its transpose?
(c) Show that Mn×n is the direct sum of the space of symmetric matrices and the space of antisymmetric matrices.