Matrix Addition¶

Definition¶

If $\boldsymbol{A}=(a_{ij})$ and $\boldsymbol{B}=(b_{ij})$ have the same size, their sum is the matrix $\boldsymbol{A}+\boldsymbol{B}=(a_{ij}+b_{ij})$$b_{ij})$.

Interpretation¶

Matrix addition works entry by entry: the entry in row $i$ and column $j$ of $\boldsymbol{A}+\boldsymbol{B}$ is $a_{ij}+b_{ij}$$b_{ij}$. That is why both matrices must have the same size: a $2\times3$ and a $3\times2$ matrix have no matching positions to pair.

Because each entry is an ordinary sum in the field, the rules of addition carry over: besides the Key Results, the zero matrix is neutral and $-\boldsymbol{A}$ cancels $\boldsymbol{A}$, so the $m\times n$ matrices form an abelian group under addition. For maps, $(\boldsymbol{A}+\boldsymbol{B})\boldsymbol{x}=\boldsymbol{A}\boldsymbol{x}+\boldsymbol{B}\boldsymbol{x}$$\mathbf{B}\mathbf{x}$, which matches the sum of linear maps. Entry-by-entry thinking does not carry over to multiplication: the matrix product is not computed entry by entry.

Key Results¶

• Matrix addition is defined only for matrices of the same size.
• Matrix addition is associative and commutative.

Examples¶

• $\begin{pmatrix}1&2\\3&4\end{pmatrix}+\begin{pmatrix}0&1\\1&0\end{pmatrix}=\begin{pmatrix}1&3\\4&4\end{pmatrix}$$\left(\begin{array}{cc}1& 3\\ 4& 4\end{array}\right)$.
• A matrix plus the zero matrix of the same size equals itself.

Prerequisites¶

• Matrix

Related Concepts¶

• Matrix Scalar Multiplication
• Vector Addition
• Vector Space
• Linear Map
• Block Matrix
• Trace

Sources¶

• Math II: Linear Algebra · Week 3

Aliases: addition of matrices, matrix sum, sum of matrices

Last updated: 20 October 2026 03:02