## Definition

Let $\map {\mathcal M_S} {m, n}$ be a $m \times n$ matrix space over $S$ over an algebraic structure $\struct {S, \circ}$.

Let $\mathbf A, \mathbf B \in \map {\mathcal M_S} {m, n}$.

Then the matrix entrywise sum of $\mathbf A$ and $\mathbf B$ (or just matrix sum) is written $\mathbf A + \mathbf B$, and is defined as follows.

Let $\mathbf A + \mathbf B = \mathbf C = \sqbrk c_{m n}$.

Then:

$\forall i \in \closedint 1 m, j \in \closedint 1 n: c_{i j} = a_{i j} \circ b_{i j}$

Thus $\sqbrk c_{m n}$ is the $m \times n$ matrix whose entries are made by performing the operation $\circ$ on corresponding entries of $\mathbf A$ and $\mathbf B$.

It follows that matrix entrywise addition is defined only when both matrices have the same number of rows and the same number of columns.

## Examples

### Addition of Real $2 \times 2$ Matrices

Let $\mathbf A = \begin {pmatrix} p & q \\ r & s \end {pmatrix}$ and $\mathbf B = \begin {pmatrix} w & x \\ y & z \end {pmatrix}$ be order $2$ square matrices over the real numbers.

Then the matrix sum of $\mathbf A$ and $\mathbf B$ is given by:

$\mathbf A + \mathbf B = \begin {pmatrix} p + w & q + x \\ r + y & s + z \end {pmatrix}$

## Also see

There are several types of addition defined on matrices.

• Matrix entrywise addition is the most common, and (at elementary level) it is just known as matrix addition

When more than one is being used during the course of an exposition, it is a very good idea to specify them with their full names whenever invoked.