On a matrix product question in cryptography

Let A,B be invertible n×n matrices over the finite field Fq with irreducible characteristic polynomials. For k∈Z+, denoteMk(A,B):={f(A)g(B):f,g∈Fq[x],with degf,degg4−kqmin(n,2k−1). Moreover, let d=dimKer(AB−BA), we prove|Mk(A,B)|>116(nd)(2(n−d))n−d2qk+min(k2,n−d2).