The title is the question. This technique involves using the "matrix of cofactors", or "adjugate matrix", and gives explicit formulae for the components of the inverse of a square matrix. It is not easy to do by hand for a matrix bigger than, say, $3\times 3$. For an $n\times n$ matrix, it requires computing the determinant of the matrix itself and computing $n^2$ determinants of $(n-1)\times(n-1)$ matrices. So I'm guessing it is not that useful for applications. But I would like confirmation.
I am not asking about the theoretical significance of the technique in proving theorems about matrices.