This came about from some lunchtime fun a couple of days ago – we had an empty whiteboard and a boardpen: it was the logical thing to do.

\textrm{let } M =\begin{pmatrix}a&b&c\\ d&e&f\\ g&h&i \end{pmatrix}.
Now produce the matrix of minors

\begin{pmatrix} ei-fh&di-fg&dh-eg\\ bi-ch&ai-cg&ah-bg\\ bf-ce&af-cd&ae-bd \end{pmatrix}.

Use the alternating law of signs to produce the matrix of cofactors M'

M' = \begin{pmatrix} ei-fh&fg-di&dh-eg\\ ch-bi&ai-cg&bg-ah\\ bf-ce&cd-af&ae-bd \end{pmatrix}

Transpose M'

{M'}^T = \begin{pmatrix} ei-fh&ch-bi&bf-ce\\ fg-di&ai-cg&cd-af\\ dh-eg&bg-ah&ae-bd \end{pmatrix}

\textrm{det }M = a(ei-fh) - b(di-fg) + c(dh-eg)

The grand formula:
M^{-1} =
\frac{1}{a(ei-fh) - b(di-fg) + c(dh-eg)} \begin{pmatrix} ei-fh&ch-bi&bf-ce\\ fg-di&ai-cg&cd-af\\ dh-eg&bg-ah&ae-bd \end{pmatrix}

I suggest you print it out and put it on your bedroom ceiling…