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)

implies

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.

𝐀+𝐁=𝐁+𝐀\mathbf{A+B=B+A}

Matrix addition is also associative.

(𝐀+𝐁)+𝐂=𝐀+(𝐁+𝐂)\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.

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