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