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

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