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Math Help - Matrix Probelm

  1. #1
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    Matrix Probelm

    Let A and B be nonsingular matrices of the same order. Show that AB is nonsigular and (AB)^-1 = B^-1 * A^-1

    Hint: Multiply by AB


    Any advice...I don't know how to show this

    Thanks.
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  2. #2
    MHF Contributor
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    Quote Originally Posted by jzellt View Post
    Let A and B be nonsingular matrices of the same order. Show that AB is nonsigular and (AB)^-1 = B^-1 * A^-1

    Hint: Multiply by AB


    Any advice...I don't know how to show this

    Thanks.
    HI

    Let C=AB

    C(B^{-1}A^{-1})=AB(B^{-1}A^{-1})

    =A(BB^{-1})A^{-1}

    =AIA^{-1}

    =AA^{-1}

    =I

    Similarly , (B^{-1}A^{-1})C=(B^{-1}A^{-1})(AB)

    =B^{-1}(AA^{-1})B

    =I

    Since (B^{-1}A^{-1})C=C(B^{-1}A^{-1})

    For both cases , C^{-1}=B^{-1}A^{-1}

     <br />
(AB)^{-1}=B^{-1}A^{-1}<br />
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  3. #3
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    Determinant of Matrix Product

    Hello jzellt
    Quote Originally Posted by jzellt View Post
    Let A and B be nonsingular matrices of the same order. Show that AB is nonsigular and (AB)^-1 = B^-1 * A^-1

    Hint: Multiply by AB


    Any advice...I don't know how to show this

    Thanks.
    mathaddict has given you the proof that (AB)^{-1}=B^{-1}A^{-1}.

    The first part of the proof (that AB is non-singular) is very straightforward:
    A and B are non-singular

    \Rightarrow\det(A)\ne0 and \det(B)\ne0

    \Rightarrow \det(A)\det(B) \ne 0

    But  \det(A)\det(B)=\det(AB)

    \Rightarrow \det(AB)\ne0

    \Rightarrow AB is non-singular
    Grandad
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