Verify the identities (assume all indicated operations are defined)

Question

a) (A^−1+B^−1)^−1=A(A+B)^−1B

b) (A+BB^T)^−1B =A^−1B(I +B^TA^−1B)^−1

Answer 1

I'll do part (a).

We only need to check A^-1+B^-1 is a one-sided inverse of the RHS. So the following calculation is enough:

(A^-1+B^-1)(A(A+B)^-1B)

= A^-1A(A+B)^-1B+B^-1A(A+B)^-1B

= (A+B)^-1B+B^-1A(A+B)^-1B

= (I+B^-1A)(A+B)^-1B

= B^-1(A+B)(A+B)^-1B

= I