Diophantine property of matrices and attractors of projective iterated function systems in ${\mathbb{R}\mathbb{P}}^1$

We prove that almost every finite collection of matrices in $G{L}_d(\mathbb{R})$ and $S{L}_d(\mathbb{R})$ with positive entries is Diophantine. Next we restrict ourselves to the case $d=2$. A finite set of $S{L}_2(\mathbb{R})$ matrices induces a (generalized) iterated function system on the projective line ${\mathbb{R}\mathbb{P}}^1$. Assuming uniform hyperbolicity and the Diophantine property, we show that the dimension of the attractor equals the minimum of 1 and the critical exponent.