Q:

Proving facts about the inverse of a matrix

Let A and B be matrices. Show that:
$(A^{-1})^{-1} = A$
$(A^{T})^{-1} = (A^{-1})^{T}$
$(AB)^{-1} = B^{-1}A^{-1}$
I think I'm supposed to use the inverse property (That $AA^{-1} = I$, where I is the identity matrix.), but I'm struggling to prove the last two. The first pretty much follows in from the definition of the inverse.

A:

(2) $$(A^{-1})^TA^T=(AA^{-1})^T=I^T=I$$
$$A^T(A^{-1})^T=(A^{-1}A)^T=I^T=I$$
(3)$$(B^{-1}A^{-1})(AB)=B^{-1}(A^{-1}A)B=B^{-1}IB=B^{-1}B=I$$
$$(AB)(B^{-1}A^{-1})=A(BB^{-1})A^{-1}=AIA^{-1}=AA^{-1}=I$$

