Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Is there a proof for the following identity?


share|cite|improve this question
what happens if $a=i = -b?$ –  abel May 1 '11 at 23:25
Well, I generously assumed that everything is defined. –  Phira May 1 '11 at 23:27

3 Answers 3

up vote 8 down vote accepted

$$ \left( B^{-1}A + I\right) = \left(I + B^{-1}A\right) $$

$$ B^{-1} \left( A +B \right) = \left(A^{-1} + B^{-1}\right) A $$

$$ \left(A^{-1} + B^{-1}\right)^{-1} B^{-1} = A \left( A +B \right)^{-1}$$

$$ \left(A^{-1} + B^{-1}\right)^{-1} = A \left( A +B \right)^{-1} B$$

share|cite|improve this answer

\begin{equation} (A^{-1} + B^{-1})^{-1} = \left(B^{-1}(A + B) A^{-1}\right)^{-1} \end{equation}

Since $(CD)^{-1} = D^{-1} C^{-1}$, we have

\begin{equation} \left(B^{-1}(A + B) A^{-1}\right)^{-1} = A(A+B)^{-1}B \end{equation}

share|cite|improve this answer

Write the matrices as $(A^{-1})^{-1}$ and repeatedly use $(M N)^{-1}=N^{-1}M^{-1}$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.