Proving facts about the inverse of a matrix

Question

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.

Answer 1

(2) $$(A^{-1})^TA^T=(AA^{-1})^T=I^T=I$$ $$A^T(A^{-1})^T=(A^{-1}A)^T=I^T=I$$

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

Answer 2

For the last one, let $$C=AB\Rightarrow CB^{-1}=ABB^{-1}=A\Rightarrow CB^{-1}A^{-1}=I$$

Thus $$(AB)^{-1} = B^{-1}A^{-1}$$

Answer 3

Hint: Show that

• $(A^{-1})^T A^T = I$
• $(AB)(B^{-1}A^{-1}) = I$

Why do these facts answer your question?