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In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together.

For a vector, $\overrightarrow{v}$, adding two matrices would have the geometric effect of applying each matrix transformation separately onto $\overrightarrow{v}$, then adding the transformed vectors.

$𝐀\overrightarrow{v}+𝐁\overrightarrow{v}=(𝐀+𝐁)\overrightarrow{v}$

Two matrices must have an equal number of rows and columns to be added.^[1] In which case, the sum of two matrices A and B will be a matrix which has the same number of rows and columns as A and B. The sum of A and B, denoted A + B, is computed by adding corresponding elements of A and B:^[2][3]

$\begin{array}{cc}𝐀+𝐁& =\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \cdots & a_{mn}\end{array}\right]+\left[\begin{array}{cccc}{b}_{11}& b_{12}& \cdots & b_{1n}\\ b_{21}& b_{22}& \cdots & b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ b_{m1}& b_{m2}& \cdots & b_{mn}\end{array}\right]\\ & =\left[\begin{array}{cccc}{a}_{11}+b_{11}& a_{12}+b_{12}& \cdots & a_{1n}+b_{1n}\\ a_{21}+b_{21}& a_{22}+b_{22}& \cdots & a_{2n}+b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}+b_{m1}& a_{m2}+b_{m2}& \cdots & a_{mn}+b_{mn}\end{array}\right]\end{array}$

Or more concisely (assuming that A + B = C):^[4]

$c_{ij}=a_{ij}+b_{ij}$

For example:

$\left[\begin{array}{cc}1& 3\\ 1& 0\\ 1& 2\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 7& 5\\ 2& 1\end{array}\right]=\left[\begin{array}{cc}1+0& 3+0\\ 1+7& 0+5\\ 1+2& 2+1\end{array}\right]=\left[\begin{array}{cc}1& 3\\ 8& 5\\ 3& 3\end{array}\right]$

Similarly, it is also possible to subtract one matrix from another, as long as they have the same dimensions. The difference of A and B, denoted A − B, is computed by subtracting elements of B from corresponding elements of A, and has the same dimensions as A and B. For example:

$\left[\begin{array}{cc}1& 3\\ 1& 0\\ 1& 2\end{array}\right]-\left[\begin{array}{cc}0& 0\\ 7& 5\\ 2& 1\end{array}\right]=\left[\begin{array}{cc}1-0& 3-0\\ 1-7& 0-5\\ 1-2& 2-1\end{array}\right]=\left[\begin{array}{cc}1& 3\\ -6& -5\\ -1& 1\end{array}\right]$

• Matrix multiplication
• Vector addition
• Direct sum of matrices
• Kronecker sum

• Lipschutz, Seymour; Lipson, Marc (2017). Schaum's Outline of Linear Algebra (6 ed.). McGraw-Hill Education. ISBN 9781260011449.
• Riley, K.F.; Hobson, M.P.; Bence, S.J. (2006). Mathematical methods for physics and engineering (3 ed.). Cambridge University Press. doi:10.1017/CBO9780511810763. ISBN 978-0-521-86153-3.