Suppose $AB=0$ for some non-zero matrix $B$. Can $A$ be invertible

Question

"Suppose $AB=0$ for some non-zero matrix $B$. Can $A$ be invertible?"

I think $A$ is not invertible by proof by contradiction. Assume $A$ is invertible, then $A$ has a left inverse. Multiplying both sides we get $B= (A^{-1})(0)$. Now if the right side is compatible then it should be $0$ and we get a contradiction because $B$ is suppose to be a nonzero matrix. However, because the dimensions of the matrix $A$ and $B$ are not given, I'm not sure if this works. The zero matrix may not be matrix multiplication compatible with $A$ inverse.

However, this does tell us that in the case where $A$ and $B$ are square matrix, $A$ is not invertible if $AB=0$ where $B$ is nonzero.

Now what do we do about the cases where $A$ and $B$ are not square matrices?

Answer 1

Since $A$ is invertible it is a square matrix. Now each column $y$ of $B$ satisfies $Ay=0$ which impllies $y=0$. So $B$ has to be the zero matrix.