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Math Help - invertible matrices

  1. #1
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    Exclamation invertible matrices

    Please help!

    Suppose A,B,& X are n by n matrices, with A, X, & A-AX invertable, & suppose:

    (A-AX)^-1=X^-1*B

    I know B is invertable

    But I can't solve for X.

    Any Help
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  2. #2
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    Suppose (A-AX)^-1=X^-1*B.
    Multiply on the left by X:
    X(A-AX)^-1 = B.
    Multiply on the right by (A-AX):
    X = B(A-AX).
    X + BAX = BA
    X(I+BA) = BA.
    (X-I)(I+BA) = BA - (I+BA) = -I.
    Assume I+BA is invertible.
    X-I = -(I+BA)^-1
    X = I - (I+BA)^-1.
    If I+BA is not invertible there's no solution in X.
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