Matrix Addition
Matrix addition is performed by adding the corresponding entries of two matrices that have the same dimensions.
Formula
(A + B)_{ij} = a_{ij} + b_{ij}
Definition
To add two matrices, you add the numbers in the same position from each: the two matrices must be the same size. Formally, for two $m \times n$ matrices $A$ and $B$, their sum $C = A + B$ has entries $c_{ij} = a_{ij} + b_{ij}$; addition is commutative and associative, and the zero matrix (all entries $0$) is the additive identity, $A + O = A$. This gives $M_{m,n}(F)$ the structure of an abelian group (and vector space over $F$); combined with scalar multiplication and, for square matrices, matrix multiplication, $M_n(F)$ forms a unital ring and $F$-algebra, whose Lie algebra $\mathfrak{gl}_n(F) = (M_n(F), [A,B] = AB - BA)$ uses only addition and the commutator.
Example
$\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix} + \begin{bmatrix}5 & 6\\7 & 8\end{bmatrix} = \begin{bmatrix}6 & 8\\10 & 12\end{bmatrix}$, adding entry by entry in the same spot. For $A = \begin{bmatrix}2 & -1 & 3\\0 & 4 & -2\end{bmatrix}$ and $B = \begin{bmatrix}-1 & 2 & 0\\3 & -3 & 1\end{bmatrix}$: $A + B = \begin{bmatrix}1 & 1 & 3\\3 & 1 & -1\end{bmatrix}$. The commutator $[A,B] = AB - BA$ measures how much $A$ and $B$ fail to commute, the Lie bracket central to quantum mechanics, $[\hat{x}, \hat{p}] = i\hbar I$, the Heisenberg commutation relation.
Key Insight
If the matrices have different sizes, you cannot add them, there is no matching entry for the leftover rows or columns, and the set of all $m \times n$ matrices with addition and scalar multiplication forms a vector space, meaning all the tools of linear algebra apply to matrices themselves as vectors. Matrix addition is entry-wise, but the important structure comes from its interaction with matrix multiplication, and this combination of additive and multiplicative structure makes matrix algebras a key setting for representation theory and quantum mechanics.