Let A and B be NxN matrices, and assume that their product C = AB is invertible. Without using determinants, prove that A and B must both be invertible.
If a NXN matrix A is invertible:
Ax = 0 has only the trivial solution 0.
The Attempt at a Solution
I believe I have this halfway figured out:
Let's take B to be singular, then there exists an x that is not 0 such that Bx = 0. Thus:
C = AB => Cx = (AB)x = A(Bx) = 0.
C would have to be singular, as there exists an x not equal to 0 that makes Cx = 0. This violates the assumption that C is invertible, thus B must be invertible.
Now, I am not sure where to go with A. I've tried applying the same logic, however:
C = AB => Cx = (AB)x = (Ax)B = 0
does not seem to work. Is there some easy way to prove that if B is invertible then A must be also? If not, how can I alter this logic to prove that A is invertible? Can I somehow use the fact that the product of two invertible matrices must be invertible...and if A is not invertible then C cannot be?