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let $A,B,C$ be $n\times n$ matrices with real entries such that $A$ is invertible. if $(A-B)CA=B$ show that $AC(A-B)=B$.

any Ideas??

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Let $Q = A-B$. Since $QCA=B = A-Q$, $Q(CA+I)=A$, and therefore $Q$ is invertible. Now $C = Q^{-1} B A^{-1} = Q^{-1}(A-Q)A^{-1} = Q^{-1} - A^{-1}$, so $AC(A-B) = A(Q^{-1} - A^{-1}) Q = A - Q = B$.

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    $A$ was assumed invertible. $Q(CA+I)A^{-1} = I$. Any $n \times n$ matrix with a one-sided inverse has rank $n$ and therefore is inverible.2012-02-08