# Math Help - proving that is A commutes, then its inverse does as well

1. ## proving that is A commutes, then its inverse does as well

If A and B are both invertible nxn matrices and A is invertible,

How can I prove thatA^-1 B=BA^-1

2. Originally Posted by cottekr
If A and B are both invertible nxn matrices and A is invertible,

How can I prove thatA^-1 B=BA^-1
Since they're square matrices and A is invertible, we see that

\begin{aligned}AB=BA & \implies A^{-1}\left(AB\right)A^{-1}=A^{-1}\left(BA\right)A^{-1}\\ &\implies \left(A^{-1}A\right)BA^{-1}=A^{-1}B\left(AA^{-1}\right)\\ &\implies BA^{-1}=A^{-1}B\\ &\implies A^{-1}B=BA^{-1}\end{aligned}

Does this help?