It's a valid way to test whether a (square) matrix has an inverse, but computing big determinants is a lot of effort (both from a "student work" perspective and

It's a valid way to test whether a (square) matrix has an inverse, but computing big determinants is a lot of effort (both from a "student work" perspective and from a "computational cost" perspective). The point is that in many cases, there are enormously more efficient ways to find an inverse (or, more generally, to solve matrix problems with inverses in them).