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# Addition

Put content here**Definition:** The **sum** of two matrices $A$ and $B$ of the same size ($m \times n$) is the matrix $A + B$ defined by:
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$$(A + B)_{ij} = a_{ij} + b_{ij}$$
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Matrix addition is performed **entrywise**.
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**Example:**
$$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix}$$
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**Properties:**
- Commutative: $A + B = B + A$
- Associative: $(A + B) + C = A + (B + C)$
- Identity: $A + 0 = A$
- Inverse: $A + (-A) = 0$
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These properties make the set of $m \times n$ matrices an **abelian group** under addition.

# Parents

* Operations on matrices