Show that if $A$, $B$, and $A + B$ are invertible matrices with the same size, then $A(A^{-1}+B^{-1})B(A+B)^{-1}=I$
What does the result in the first part tell you about the matrix $(A^{-1}+B^{-1})$?
I get the first part. Help me with the second part.
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$\begingroup$It tells $A^{-1}+B^{-1}$ is invertible and its inverse is $B(A+B)^{-1}A$.
$\endgroup$ 0 $\begingroup$For two $n \times n$ matrices $A$ and $B$ we find the relation $(A+B) = A(A^{-1}+B^{-1})B$ (since $(A+B)(A+B)^{-1} = I_n$). It is given that $A$ and $B$ are invertible so first premultiply this relation with $A^{-1}$ and then post multiply with $B^{-1}$ to arrive at an expression for $(A^{-1} + B^{-1})$.
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