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Math Help - If A is invertible then A+B and I+BA^-1 are both invertible or both non-invertible

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    If A is invertible then A+B and I+BA^-1 are both invertible or both non-invertible

    Prove If $A$ is invertible then $A+B$ and $I+BA^{-1}$ are both invertible or both non-invertible


    If $A+B$ is invertible, i.e. $A+B=C$ where $CC^{-1} = I$ then $(A+B)A^ {-1} = I +BA^{-1}$ but I get stuck here, I don't know how I show this leads to an invertible matrix, say D

    Edit: $A+B=C \Rightarrow (A+B)A^{-1} = CA^{-1} = I + BA^{-1} = CA^{-1} \Rightarrow I + BA^{-1} - I = CA^{-1} - I \Rightarrow BA^{-1} = CA^{-1} - I$ $\Rightarrow BA^{-1}A = (CA^{-1} - I)A \Rightarrow B = C-A \Rightarrow B+A = C-A + A \Rightarrow B+A = C$

    and if $A+B$ is not invertible then would the above argument still be used? I'm never counting on B or C to be invertible or non-invertible in that argument
    Last edited by Jonroberts74; September 1st 2014 at 06:31 PM.
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    Re: If A is invertible then A+B and I+BA^-1 are both invertible or both non-invertibl

    \begin{array}{cc}A+B = C \\ (A+B)A^{-1} = CA^{-1} \\ I +BA^{-1}=CA^{-1} \\ I +BA^{-1} - I =CA^{-1} - I \\  BA^{-1}  =CA^{-1} - I \\ (BA^{-1})A  =(CA^{-1} - I)A\\ B=C-A \\ B+A=C\end{array}

    Therefore, if $A+B$ is invertible/non-invertible then $I+BA^{-1}$ is invertible/non-invertible
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    Re: If A is invertible then A+B and I+BA^-1 are both invertible or both non-invertibl

    Consider

    A =\left(\begin{array}{cc} 2 & 1\\ 1 & 1 \end{array}\right) and B =\left(\begin{array}{cc}1 & 1\\ 2 & 1 \end{array}\right).
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    Re: If A is invertible then A+B and I+BA^-1 are both invertible or both non-invertibl

    Use the determinant condition: for any square matrix A, A is invertible iff det(A) is not equal to 0:

    $$A+B=(I+BA^{-1})A$$
    So
    $$\det(A+B)=\det((I+BA^{-1})A)=\det(I+BA^{-1})\det(A)$$
    Since $\det(A)\neq 0$,
    $$\det(A+B)\neq 0\text{ iff } \det(I+BA^{-1})\neq 0$$
    Thanks from Jonroberts74
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    Re: If A is invertible then A+B and I+BA^-1 are both invertible or both non-invertibl

    Quote Originally Posted by Jester View Post
    Consider

    A =\left(\begin{array}{cc} 2 & 1\\ 1 & 1 \end{array}\right) and B =\left(\begin{array}{cc}1 & 1\\ 2 & 1 \end{array}\right).
    Perhaps you did not understand the question. The A and B you give are invertible matrices. A+ B is not and I+ BA^-1 is not either, just as the "theorem" says. This is an example for which the statement is true but an example doesn't prove anything.
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    Re: If A is invertible then A+B and I+BA^-1 are both invertible or both non-invertibl

    I certainly did misunderstand the question. I think johng provided a nice proof.
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    Re: If A is invertible then A+B and I+BA^-1 are both invertible or both non-invertibl

    Quote Originally Posted by Jester View Post
    Consider

    A =\left(\begin{array}{cc} 2 & 1\\ 1 & 1 \end{array}\right) and B =\left(\begin{array}{cc}1 & 1\\ 2 & 1 \end{array}\right).
    $A$ has an inverse, $A+B$ has no inverse and $I+BA^{-1}$ has no inverse, so that show's its true for that that A and B

    EDIT: Didn't see the above replies

    thanks
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    Re: If A is invertible then A+B and I+BA^-1 are both invertible or both non-invertibl

    Quote Originally Posted by Jester View Post
    I certainly did misunderstand the question. I think johng provided a nice proof.
    So did I at first!
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