# Why must the product of two invertible matrices be invertible?

If $A$ has inverse ${A}^{- 1}$ and $B$ has inverse ${B}^{- 1}$, then $A B$ has inverse ${B}^{- 1} {A}^{- 1}$
$\left(A B\right) \left({B}^{- 1} {A}^{- 1}\right) = A \left(B {B}^{- 1}\right) {A}^{- 1} = A I {A}^{- 1} = A {A}^{- 1} = I$
$\left({B}^{- 1} {A}^{- 1}\right) \left(A B\right) = {B}^{- 1} \left({A}^{- 1} A\right) B = {B}^{- 1} I B = {B}^{- 1} B = I$