Matrix A,B,C 2/3 not invertible

Question

Let A,B,C be matrices such that the algebraic operations are defined.

Question: Disprove the following statement (by giving a counter example):

If AB=C and 2 of the 3 matrices are not invertible, then the third is not either

My struggle: I can't think of such an example, could I maybe get a hint in the right direction? If you prefer, you can also comment the answer to it. Thanks in advance

Answer 1

Let consider

$$A=\begin{bmatrix}1&0\\0&0\end{bmatrix}\:, B=\begin{bmatrix}1&0\\0&1\end{bmatrix}\:,C=\begin{bmatrix}1&0\\0&0\end{bmatrix}$$

Note also that since

$$\det(AB)=\det A\:\det B$$

it is impossible that $A$ and $B$ are not invertible and $C$ is invertible.