Understanding matrix inversion via the adjugate

Question

Let's say we have matrix ${\bf A}$. Its inverse matrix is given by

$$ {\bf A}^{-1} = \frac{1}{\det({\bf A})} \operatorname{adj}({\bf A}) $$

I know that dividing by the determinant makes sense because we are getting rid of the area done by the transformation but what is going on with the adjoint matrix?

Answer 1

So we know that $AA^{-1}=I$. Also, $A[\operatorname{adj}(A)] = [\det(A)]I$ (see here).

Therefore,

$$A \frac{\operatorname{adj}(A)}{\det(A)} = I \implies \frac{\operatorname{adj}(A)}{\det(A)} = A^{-1}$$