If we let $A$ be a $m\times n$ matrix and let $B$ be a $n\times p$ matrix. Suppose that both $A$ and $B$ have a left inverse. Show that $AB$ has a left inverse.
i know that matrix $A^T$A is an invertible $n$ by $n$ symmetric matrix. Hence. $(A^T)^{-1}A^TA = I$. So then $A_{\text{left}}^{-1} = (A^TA)^{-1}A^T$
But how do I go about proving that $AB$ has a left inverse?
Hint: Try computing $(B_{\text{left}}^{-1}A_{\text{left}}^{-1})(AB)$. What do you get?