Let $S$ and $T$ be two matrices. Assume that $ST$ is invertible. I want to prove that $S$ and $T$ are invertible.
I managed to prove that $T$ is invertible. Here is my proof. Assume $T(X)=0$. Then $X=I X=(ST)^{-1}(ST)X=(ST)^{-1}S(0)=0$. So $T$ is invertible.
How to prove that $S$ is also invertible ?