Linear algebra: help with proof of equality In an exercise, I need to show that
$$(I_n - A)^{-1} + (I_n - A^{-1})^{-1} = I_n + 2 (I_n - A)^{-1}A$$
What I've tried:
$$(I_n - A)^{-1} + (I_n - A^{-1})^{-1} = I_n + 2 (I_n - A)^{-1}A$$
$$\iff(I_n - A)[(I_n - A)^{-1} + (I_n - A^{-1})^{-1}](I_n - A^{-1})=(I_n - A)[I_n + 2 (I_n - A)^{-1}A](I_n - A^{-1})$$
$$\iff [I_n-(I_n-A)(I_n-A^{-1})^{-1}](I_n-A^{-1})=I_n-A+[2(I_n-A)(I_n-A)^{-1}A)(I_n-A^{-1})]$$
$$\iff I_n-A^{-1}+I_n-A=(I_n-A^{-1})(I_n+A)$$
$$\iff 2I_n-A-A^{-1}=-A^{-1}+A$$
$$\iff 2I_n=2A$$
$$\iff A=I_n$$
Which is obviously not what I had to prove. Any help with this would be appreciated :)

Actually $(I_n - A)[(I_n - A)^{-1} + (I_n - A^{-1})^{-1}](I_n - A^{-1})=I_n-A^{-1} + I_n -A$
