Determine whether the given set of matrices under the specified operation, matrix addition or multiplication, is a group. Recall that a diagonal matrix is a square matrix whose only nonzero entries lie on the main diagonal, from the upper left to the lower right comer. An upper-triangular matrix is a square matrix with only zero entries below the main diagonal. Associated with each n x n matrix A is a number called the determinant of A, denoted by det(A). If A and B are both n x n matrices, then det(AB) = det(A) det(B). Also, det(I_n) = 1 and A is invertible if and only if det(A) ≠ 0.

All n x n diagonal matrices with no zero diagonal entry under matrix multiplication.