Abstract
We study equilibrium measures (K ̈aenm ̈aki measures) supported on self-affine sets generated by a finite collection of diagonal and anti-diagonal matrices acting on the plane and satisfying the strong separation property. Our main result is that such measures are exact dimensional and the dimension satisfies the Ledrappier-Young formula, which gives an explicit expression for the dimension in terms of the entropy and Lyapunov exponents as well as the dimension of a coordinate projection of the measure. In particular, we do this by showing that the K ̈aenm ̈aki measure is equal to the sum of (the pushforwards) of two Gibbs measures on an associated subshift of finite type.