Originally Posted by

**alunw** A linear transformation is invertible if and only if its matrix has a non-zero determinant. It is surely easier to calculate the determinant than the inverse, so this is a sensible l thing to do. The determinant is the measure of the transformed unit "hypercube", so is non-zero if and only if the kernel is trivial.

On an even more practical level the distinction between "invertible" and "non-invertible" is not so clear as it is in theory. If you try to calculate the determinant of a matrix and get an answer very close to 0 but not exactly 0 you may well not really be sure if you got a non-zero answer because of round-off errors in the calculation or in the entries in the matrix itself.