I'm having trouble proving this without using determinants. I know how to prove it with the product of just two matrices, but I'm not sure how to generalize this to a product of k matrices. Is there a way to do this proof without determinants?
To clarify the question, each matrix is an $n$ by $n$ matrix.
For example, I know that if the determinant of the product is nonzero, then all of the determinants of the individual matrices must also be nonzero.
I also know that if a product of two matrices $A$ and $B$, $AB$ is nonsingular, then there exists a matrix $C$ so that $C(AB) = I$ and $(AB)C = I$, and so $(CA)B = I$ and $A(BC) = I$, so $A$ and $B$ are both invertible, and thus nonsingular.
I'm looking for a way to generalize this. Or just any other way to prove this without determinants.
Thanks so much!