These matrices occur quite frequently in applications, so it is worthwhile to study some of their most important properties.

(a) Verify the claims in (11) that α_kj = α_jk for a symmetric matrix, and α_kj = -α_jk for a skew symmetric matrix. Give examples.
(b) Show that for every square matrix C the matrix C + C^T is symmetric and C- C^T is skew-symmetric. Write C in the form C = S + T, where S is symmetric and T is skew-symmetric and find S and T in terms of C. Represent A and B in Probs. 11–20 in this form.

(c) A linear combination of matrices A, B, C,· · · ·, M of the same size is an expression of the form

(14) αA + bB + cC + · · · · +mM,

where α,· · ·, m are any scalars. Show that if these matrices are square and symmetric, so is (14); similarly, if they are skew-symmetric, so is (14).

(d) Show that AB with symmetric A and B is symmetric if and only if A and B commute, that is, AB = BA.

(e) Under what condition is the product of skew-symmetric matrices skew-symmetric?