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Thread: 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
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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 $\displaystyle C=AB$

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

    $\displaystyle =A(BB^{-1})A^{-1}$

    $\displaystyle =AIA^{-1}$

    $\displaystyle =AA^{-1}$

    $\displaystyle =I$

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

    $\displaystyle =B^{-1}(AA^{-1})B$

    $\displaystyle =I$

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

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

    $\displaystyle
    (AB)^{-1}=B^{-1}A^{-1}
    $
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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 $\displaystyle (AB)^{-1}=B^{-1}A^{-1}$.

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

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

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

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

    $\displaystyle \Rightarrow \det(AB)\ne0$

    $\displaystyle \Rightarrow AB$ is non-singular
    Grandad
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