I will describe joint work with Boris Solomyak, in which we show that the stationary (Furstenberg) measure on the projective line associated to 2x2 random matrix products has the "correct" dimension (entropy / Lyapunov exponent) provided that the matrices satisfy a certain diophantine property, which is satisfied e.g. by all matrices with algebraic entries. Together with results of Breuilard and Gelander on uniform expansion in linear groups, this implies that there is a neighborhood $U$ of the identity matrix such that for every pair of matrices in $U$ with algebraic entries, the stationary measure has full dimension $(\dim = 1)$.