What is the intution behind the matrix inverse formula? I am learning about matrix inverses and my professor introduced the formula $\begin{bmatrix}a & b\\c & d\end{bmatrix}^{-1}=$ $ 1\over(ad-bc)$$\begin{bmatrix}d & -b\\-c & a\end{bmatrix}$ $,if\; and \; only \; if \; ad-bc\neq
0$. Can someone give me some intuition as to why this formula works?
I understand the intuition behind using Gauss-Jordan elimination to find the matrix by applying all of the steps of row reduction to the identity matrix, but this formula seems like magic.
Thank you!

Inverse of an invertible matrix is given by
$$A^{-1}={\frac {1}{\det(A)}} \mathrm{adj} (A)$$
Clearly, $\det(A) = ad-bc$ and we can find the adjugate matrix by applying transpose on the cofactor matrix of $A$. This results in the formula
$$A ^{-1} = {\frac {1}{ad-bc}} {\begin{bmatrix}\,\,\,d&\!\!-b\\-c&\,a\\\end{bmatrix}}$$
