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