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) |

Matrix addition is undefined unless the addends have the same dimensions. Matrix addition is commutative.

$\mathbf{A+B=B+A}$ |

Matrix addition is also associative.

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

The additive identity is the zero matrix. The additive inverse of matrix A is denoted by -A and consists of the element by element negation of a A, i.e. it’s the matrix formed when a A is multiplied by the scalar -1.

$\mathbf{-A}=-1\cdot\mathbf{A}$ | (11) |