Playing around with my CAS, I found that apparently

$$\sum\_{n=1}^\infty\begin{bmatrix}1/2 & 1/3\\1/4 & 1/5\end{bmatrix}^n = \begin{bmatrix}29/19 & 20/19\\15/19 & 11/19\end{bmatrix}$$

If that is indeed true, it's pretty cool.

How can I prove this, and what is the general formula if the coefficients are replaced by arbitrary numbers $a,b,c,d<1$?