Proof of Theorem SS

At the risk of carrying our everyday analogies too far, the proof of this theorem is quite easy when we compare it to the workings of a dating service. We have a statement about the inverse of the matrix $AB$, which for all we know right now might not even exist. Suppose $AB$ was to sign up for a dating service with two requirements for a compatible date. Upon multiplication on the left, and on the right, the result should be the identity matrix. In other words, $AB$'s ideal date would be its inverse.

Now along comes the matrix $\inverse{B}\inverse{A}$ (which we know exists because our hypothesis says both $A$ and $B$ are invertible and we can form the product of these two matrices), also looking for a date. Let us see if $\inverse{B}\inverse{A}$ is a good match for $AB$. First they meet at a noncommittal neutral location, say a coffee shop, for quiet conversation: \begin{align*} (\inverse{B}\inverse{A})(AB) &=\inverse{B}(\inverse{A}A)B&&\knowl{./knowls/theorem.MMA.knowl}{\text{Theorem MMA}}\\ &=\inverse{B}I_nB&&\knowl{./knowls/definition.MI.knowl}{\text{Definition MI}}\\ &=\inverse{B}B&&\knowl{./knowls/theorem.MMIM.knowl}{\text{Theorem MMIM}}\\ &=I_n&&\knowl{./knowls/definition.MI.knowl}{\text{Definition MI}} \end{align*} The first date having gone smoothly, a second, more serious, date is arranged, say dinner and a show: \begin{align*} (AB)(\inverse{B}\inverse{A}) &=A(B\inverse{B})\inverse{A}&&\knowl{./knowls/theorem.MMA.knowl}{\text{Theorem MMA}}\\ &=AI_n\inverse{A}&&\knowl{./knowls/definition.MI.knowl}{\text{Definition MI}}\\ &=A\inverse{A}&&\knowl{./knowls/theorem.MMIM.knowl}{\text{Theorem MMIM}}\\ &=I_n&&\knowl{./knowls/definition.MI.knowl}{\text{Definition MI}} \end{align*}

So the matrix $\inverse{B}\inverse{A}$ has met all of the requirements to be $AB$'s inverse (date) and with the ensuing marriage proposal we can announce that $\inverse{(AB)}=\inverse{B}\inverse{A}$.