Let A B and C be NxM matrices and X_{i,j} denote the i,j th elements of a matrix C. As the i j th component of the sum of two matrices is the sum

of their individual i j th components and ordinary addition is associative:

Then [(A+B)+C]_{i,j} = (A+B)_{i,j} + C_{i,j} = A_{i,j} + B_{i,j} + C_{i,j}

.............................= A_{i,j} + (B+C)_{i,j} = [A+(B+C)]_{i,j}

RonL