What is known about matrix multiplication, and matrix circuits?
While I don't know about the complexity of matrix polynomials in general, the state of the art in the complexity of matrix multiplication is nicely summarized in a recent ToC Graduate Survey by Markus Bläser.

Computing entries mod $p$ is basically the same as the question of the complexity of performing the computation over the field $\mathbb{F}_p$, which is fairly well-studied (though perhaps not as well-studied as characteristic zero). 

Most matrix multiplication research works in the nonuniform setting anyways, so it should almost all be relevant to the question of circuit size.