Let $μ$ be a measure on $SL_{2}(\mathbb{R})$ generating a non-compact and totally irreducible subgroup, let $χ>0$ denote its Lyapunov exponent, and let $ν$ be the associated stationary (Furstenberg) measure for the action on the projective line. We prove that if $μ$ is supported on finitely many matrices with algebraic entries, then \[ \dimν=\min\{1,\frac{h_{\textrm{RW}}(μ)}{2χ}\} \] where $h_{\textrm{RW}}(μ)$ is the random walk entropy of $μ$, and $\dim$ denotes pointwise dimension. In particular, for every $δ>0$, there is a neighborhood $U$ of the identity in $SL_{2}(\mathbb{R})$ such that if a measure $μ\in\mathcal{P}(U)$ is supported on algebraic matrices with all atoms of size at least $δ$, and generates a group which is non-compact and totally irreducible, then its stationary measure $ν$ satisfies $\dimν=1$.