The sum of m × n matrices A and B is an m × n matrix C which is the element by element sum of the addends.

 $\mathbf{C=A+B}$ (9)

implies

 $c_{ij}=a_{ij}+b_{ij},\operatorname{where}1\leq i\leq m\operatorname{and}1\leq j\leq n$ (10)

 $\mathbf{A+B=B+A}$
 $\mathbf{(A+B)+C=A+(B+C)}$
 $\mathbf{-A}=-1\cdot\mathbf{A}$ (11)