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Show that $P^{-1}AP=B$ where $P=(I-(A+B))$.Given that $A^2=A$ and $B^2=B$. .Please explain this step by step since I am unable to solve this problem .I only solve $A(I-(A+B))=A-A^2-AB=-AB$ since $A^2=A$.

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    Please, first, explain what you have tried so far.2012-08-02
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    Write $AP = PB$ and see what that gives you. It's just basic algebra.2012-08-02
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    @KarolisJuodelė: thanks. You solve my problem2012-08-02
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    Note that $P$ might not be invertible, so, to be precise, you can only show that $P^{-1}AP=B$ if $P$ is invertible (whereas $AP=PB$ is independent of that).2012-08-02

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We have $$ A(I-(A+B)) = A - A^2 - AB = -AB $$ Also we have $$ (I-(A+B))B = B - AB - B^2 = -AB $$ So we can conclude the the LHS on both equations are equal, i.e. $$ A(I-(A+B)) = (I-(A+B))B $$ which is $AP = PB,$ and if $P$ is invertible, we have $P^{-1} AP = P^{-1}PB = B.$

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    Please point out which step is confusing you?2012-08-02
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    I got it from Karolis comment but thank you for complete solution2012-08-02
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$$PB=(I-(A+B))B=B-AB-B^2=-AB\\AP=A(I-(A+B))=A-A^2-AB=-AB$$