I understand how to use Gaussian elimination to find the inverse of a square matrix, though I’m not a fan of tedious calculations by hand, especially considering the cubic time complexity of the method. So I would like to know if there are any simple and concise derivations of general inverse formulae for square matrices that do not use this algorithm. Firstly, is there a nice alternative derivation for the following closed form $2 \times 2$ matrix formula?
$$ A^{-1} = \pmatrix{a&b\cr c&d\cr}^{-1}={1\over\det A}\pmatrix{d&-b\cr-c&a\cr} $$
Can such a derivation technique generalize to $n \times n$ matrices of arbitrary dimension?