Let $A$ be a square matrix that is non-invertible. I was wondering if there is a simple proof that we can apply elementary row operations to get a zero row. (For a matrix $C$ to be invertible, I mean there is $B$ such that $CB = BC = I$.)
I can prove this using elementary column operations but I would like a more direct proof that doesn’t appeal to column operations or the fact that row rank equals column rank, or anything to do with transposes, or the existence of RREF, or determinants, etc. The difficulty seems to be that elementary row operations are applied on the row space, whereas invertibility is sort of defined in terms of the column space.
You could also use (but it is preferred not to) facts like: A matrix $C$ being invertible is equivalent to null space of $C$ being zero (i.e. injective) is equivalent to $C$ being surjective.