AbstractLet F=GF(q) denote the finite field of order q, and let ƒ(x)ϵF[x]. Then f(x) defines, via substitution, a function from Fn×n, the n×n matrices over F, to itself. Any function ƒ:Fn×n → Fn×n which can be represented by a polynomialf(x)ϵF[x] is called a scalar polynomial function on Fn×n. After first determining the number of scalar polynomial functions on Fn×n, the authors find necessary and sufficient conditions on a polynomial ƒ(x) ϵ F[x] in order that it defines a permutation of (i) Dn, the diagonalizable matrices in Fn×n, (ii)Rn, the matrices in Fn×n all of whose roots are in F, and (iii) the matric ring Fn×n itself. The results for (i) and (ii) are valid for an arbitrary field F.