Matrix addition

Definition

Let $R$ be a ring and let $A = (a_{ij})$ and $B = (b_{ij})$ be $m \times n$ matrices over $R$. The sum $A + B$ is the $m \times n$ matrix with entries $$(A + B)_{ij} = a_{ij} + b_{ij}$$ for $1 \le i \le m$ and $1 \le j \le n$.

Remarks

The sum is defined only for two matrices of the same size. Matrix addition is associative and commutative, the zero matrix is its identity element, and $-A = (-a_{ij})$ is the additive inverse of $A$. Thus $M_{m \times n}(R)$ is an abelian group under addition. For example, $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 3 \\ 4 & 4 \end{pmatrix}$.

Explanations

L1 human

To add two matrices of the same size, add the numbers in the same position. The result has the same size as the two matrices. Matrix addition has the same rules as the addition of numbers.