Find two $2\times2$ real matrix $A$ and $B$ such that $A$, $B$ , $A+B$ are all invertible with $(A+B)^{-1}=A^{-1}+B^{-1}$ Find two $2\times2$ real matrices $A$ and $B$ such that $A$, $B$ , $A+B$ are all invertible with $(A+B)^{-1}=A^{-1}+B^{-1}$

Tried to write the matrices as
$$A=\pmatrix {a&b\\c&d},B=\pmatrix {e&f\\g&h}$$
and solve $(A+B)^{-1}=A^{-1}+B^{-1}$, But make it too complex. Any more convenient ways?

You get $I = (A^{-1}+B^{-1})(A+B) = I + A^{-1}B + B^{-1}A + I$. Thus we get:
$$(A^{-1}B) + (A^{-1}B)^{-1} = -I$$
$$(A^{-1}B)^2 + I = -(A^{-1}B)$$
So $A^{-1}B$ satisfies the polynomial $x^2 + x + 1 = 0$. Take any matrix satisfying this polynomial; for example you can take 
$$A^{-1}B = \begin{bmatrix} -1 &1 \\ -1&0 \end{bmatrix}$$
$$B = A\begin{bmatrix} -1 &1 \\ -1&0 \end{bmatrix}$$
Hence you can take any invertible $A$ and produce $B$ of the wanted form.
