Constructing matrices with prescribed off-diagonal submatrix and invariant polynomials

Let F be any field and let B a matrix of Fq×p. Zaballa found necessary and sufficient conditions for the existence of a matrix A=[Aij]i,j∈{1,2}∈F(p+q)×(p+q) with prescribed similarity class and such that A21=B. For fields of characteristic different from 2, we obtain a finite step algorithm to construct A when it exists.