2.4 Matrix Addition

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


cij=aij+bij,where1imand1jnc_{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.


Matrix addition is also associative.


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