A family F of square matrices of the same order is called a quasi-commuting family if ( AB - BA ) C = C ( AB - BA ) for all A , B , C ∈ F where A , B , C need not be distinct. Let f k ( x 1 , x 2 , … , x p ) , ( k = 1 , 2 , … , r ) , be polynomials in the indeterminates x 1 , x 2 , … , x p with coefficients in the complex field C , and let M 1 , M 2 , … , M r be n × n matrices over C which are not necessarily distinct. Let F ( x 1 , x 2 , … , x p ) = ∑ k = 1 r M k f k ( x 1 , x 2 , … , x p ) and let δ F ( x 1 , x 2 , … , x p ) = det F ( x 1 , x 2 , … , x p ) . In this paper, we prove that, for n × n matrices A 1 , A 2 , … , A p over C , if { A 1 , A 2 , … , A p , M 1 , M 2 , … , M r } is a quasi-commuting family, then F ( A 1 , A 2 , … , A p ) = O implies that δ F ( A 1 , A 2 , … , A p ) = O .