12. Simplify each expression below, assuming that
all matrices are invertible and of the same size.
(Recall that $(AB)^{-1} = B^{-1}A^{-1}$ and $(AB)^T = B^T A^T.)$
(a) $(AC^{-1})^{-1}(AC^{-1})(AC^{-1})^{-1}AD^{-1}$
(b) $(A^{-1}(B^T + A^T)^T - I)(AB)^{-1}$
(c) $((A^T(BA^{-1})^T(B^{-1} + A^{-1})^T)^T$
13. (a) Show that if a square matrix A satisfies
$A^2 + 4A - I = 0$, then $A^{-1} = A + 4I$.
(b) Knowing that the square matrix A satis-
fies $A^3 - 2A - I = 0$, find its inverse.
