I was wondering if its possible to know if a matrix is invertible without row reducing it in attempt to make it I. If the matrix is not nxn, it is never invertible?
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).