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

  1. #1
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    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
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by cottekr View Post
    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}

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