Well, since you liked the last explanation, perhaps one in the same vein. Pretend for a second that a matrix really is just a linear transformation , then one only has to check any of the following equivalent conditions:
1) is injective
2) is surjective
3) does not have zero as an eigenvalue
4)
And there are many more, some of which are very similar to the ones I listed (e.g. 1) is easily seen to be eqiuvalent to the existence of a left inverse).