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?

Using the formula from 
Inverse of the sum of matrices
we can take, for arbitrary $a,b$ with $a+b\neq 0$,
$$
A=\begin{pmatrix} -1 & 1 \cr 0 & 2 \end{pmatrix},\quad 
B=\begin{pmatrix} a & b \cr \frac{2(a^2-a+1)}{a+b} &  \frac{-2(b-ab+1)}{a+b} \end{pmatrix}.
$$
All of $A$, $B$ and $A+B$ have determinant $-2$. It is easy to check that $(A+B)^{-1}=A^{-1}+B^{-1}$. So your way is not too "complex".
