# Is the product of two invertible matrices also invertible?

Oct 20, 2015

Matrix multiplication is associative, so $\left(A B\right) C = A \left(B C\right)$ and we can just write $A B C$ unambiguously.
Suppose $A$ and $B$ are invertible, with inverses ${A}^{-} 1$ and ${B}^{-} 1$. Then ${B}^{-} 1 {A}^{-} 1$ is the inverse of $A B$:
$\left(A B\right) \left({B}^{-} 1 {A}^{-} 1\right) = A B {B}^{-} 1 {A}^{-} 1 = A I {A}^{-} 1 = A {A}^{-} 1 = I$