The first two parts of this question appeared as Exercise 27.
(a) Show that (GH)T = HTGT.
(b) A square matrix is symmetric if each i, j entry equals the j, i entry (that is, if the matrix equals its transpose). Show that the matrices HHT and HTH are symmetric.
(c) Show that the inverse of the transpose is the transpose of the inverse.
(d) Show that the inverse of a symmetric matrix is symmetric.