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.

C = A + B

implies

c_{ij} = a_{ij} + b_{ij}, where 1 ≤ i ≤ m and 1 ≤ j ≤ n

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

A + B = B + A

Matrix addition is also associative.

(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.

–A = –1 · A