Let $A$, $B$, and $A + B$ be nonsingular matrices. Prove that $A^{-1} + B^{-1}$ is nonsingular by showing that
$( A^{-1} + B^{-1} )^{-1} = A( A + B )^{-1} B$ I have done progress to only knowing that $( A^{-1} + B^{-1} )^{-1} = (A + B)$ and from there am lost completely. Someone please give me pointers
To show that $A(A + B)^{-1} B$ is the inverse to $A^{-1} + B^{-1}$, you need to verify the definition of inverses. Specifically, we call the matrix $C$ the inverse of $A^{-1} + B^{-1}$ if $$C(A^{-1} + B^{-1}) = (A^{-1} + B^{-1})C = I.$$ If such a matrix $C$ exists, then it must be unique, hence why we can call it the inverse.
That is, we just need to show, $$A(A + B)^{-1} B(A^{-1} + B^{-1}) = (A^{-1} + B^{-1})A(A + B)^{-1} B = I.$$
This is a little daunting to do as written. First note that $$A(A + B)^{-1} B + A(A + B)^{-1} A = A(A + B)^{-1}(A + B) = AI = A,$$ hence $$A(A + B)^{-1} B = A - A(A + B)^{-1} A.$$ Similarly, $$A(A + B)^{-1} B = B - B(A + B)^{-1} B.$$ Hence, \begin{align*} (A^{-1} + B^{-1})A(A + B)^{-1} B &= A^{-1} (A(A + B)^{-1} B) + B^{-1} (A(A + B)^{-1} B) \\ &= A^{-1}(A - A(A + B)^{-1} A) + B^{-1}(B - B(A + B)^{-1} B) \\ &= I - (A + B)^{-1} A + I - (A + B)^{-1} B \\ &= 2I - (A + B)^{-1} (A + B) \\ &= I. \end{align*} See if you can show that $$A(A + B)^{-1} B(A^{-1} + B^{-1}) = I$$ in a similar manner!