<s>[INST] Prove that $A^{-1} + B^{-1}$ nonsingular by showing that $(A^{-1} + B^{-1} )^{-1} = A( A + B )^{-1} B$ 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
 [/INST] 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!
</s>