Matrices over finite fields of odd characteristic as sums of diagonalizable and square-zero matrices

Let F be a finite field of odd characteristic. When |F| >= 5 , we prove that every matrix A admits a decomposition into D+M , where D is diagonalizable and M-2=0. For F=F-3, we show that such a decomposition is possible for non-derogatory matrices of order at least 5, and more generally, for matrices whose first invariant factor is not a non-zero trace irreducible polynomial of degree 3; we also establish that matrices consisting of direct sums of companion matrices, all of them associated to the same irreducible polynomial of non-zero trace and degree 3 over F-3, never admit such a decomposition. These results completely settle the question posed by Breaz (2018) [3] asking if it is true that, for big enough positive integers n >= 3, all matrices A over a field of odd cardinality q admit decompositions of the form E+M with E-q = E and M-2 = 0 : specifically, the answer is yes for q >= 5 , but however there are counterexamples for q=3 and each order n=3k, whenever k >= 1.