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) |