Bäcklund transformation of matrix equations and a discrete matrix first Painlevé equation

We show that the known auto-Bäcklund transformation for the matrix second Painlevé equation can be generalized to a much wider class of equations. This auto-Bäcklund transformation is an involution and so cannot be used on its own to generate an infinite sequence of different solutions, although for particular equations a second auto-Bäcklund transformation allows this to be done. We also give a Bäcklund transformation for this general class of matrix equations. For the matrix second Painlevé equation we also give a coalescence limit, and a construction of special integrals and of a discrete matrix first Painlevé equation.