# Does this matrix identity hold?

For invertible matrices A and B does the identity:

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

hold? My supervisor suggested that they are equal but I haven't been able to prove this and in the matrix cookbook (http://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf) there are separate identities for both sides of this equation, but they are not given as equal to each other.

• Smells like The Woodbury identity. – Fabian Jan 19 '16 at 10:10
• Don't we first need to assume that $A+B$ is invertible? – Arpit Kansal Jan 19 '16 at 10:11
• @ArpitKansal There is a $(A+B)^{-1}$ in the identity. Obviously you need to assume that $A+B$ is invertible... – Najib Idrissi Jan 19 '16 at 10:15
• Page 18, relation 157 of your linked pdf. – N74 Jan 19 '16 at 10:16
• @Fabian It smells even more like Hua's identity, because it is just the second form listed at wikipedia with $a$ and $b$ exchanged with their respective inverses. I'm glad to learn of this more general thing, though.. – rschwieb Jan 19 '16 at 12:53