Suppose that both $A$ and $B$ have a left inverse. Show that $AB$ has a left inverse.

Question

Let $A$ be an $m×n$ matrix and $B$ be an $n×p$ matrix. Suppose that both $A$ and $B$ have a left inverse. Show that $AB$ has a left inverse.

So I know that $A_L^{-1}A=I$ so then $B_LB^{-1}B=I$. Therefore $(A_L^{-1}A)(B_LB^{-1}B)=I^2=I$

Looking for see if this proof is correct

Answer 1

You can write $(B_L^{-1}A_L^{-1})AB=I$, so $(B_L^{-1}A_L^{-1})$ is the left inverse of $AB$.