Addition

Definition: The sum of two matrices $A$ and $B$ of the same size ($m \times n$) is the matrix $A + B$ defined by:

$$(A + B)_{ij} = a_{ij} + b_{ij}$$

Matrix addition is performed entrywise.

Example:
$$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix}$$

Properties:

• Commutative: $A + B = B + A$
• Associative: $(A + B) + C = A + (B + C)$
• Identity: $A + 0 = A$
• Inverse: $A + (-A) = 0$

These properties make the set of $m \times n$ matrices an abelian group under addition.