Matrix addition

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

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

${\displaystyle \mathbf {A} {\vec {v}}+\mathbf {B} {\vec {v}}=(\mathbf {A} +\mathbf {B} ){\vec {v}}\!}${\displaystyle \mathbf {A} {\vec {v}}+\mathbf {B} {\vec {v}}=(\mathbf {A} +\mathbf {B} ){\vec {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]}

${\displaystyle {\begin{aligned}\mathbf {A} +\mathbf {B} &={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\\\end{bmatrix}}+{\begin{bmatrix}b_{11}&b_{12}&\cdots &b_{1n}\\b_{21}&b_{22}&\cdots &b_{2n}\\\vdots &\vdots &\ddots &\vdots \\b_{m1}&b_{m2}&\cdots &b_{mn}\\\end{bmatrix}}\\&={\begin{bmatrix}a_{11}+b_{11}&a_{12}+b_{12}&\cdots &a_{1n}+b_{1n}\\a_{21}+b_{21}&a_{22}+b_{22}&\cdots &a_{2n}+b_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}+b_{m1}&a_{m2}+b_{m2}&\cdots &a_{mn}+b_{mn}\\\end{bmatrix}}\\\end{aligned}},円\!}${\displaystyle {\begin{aligned}\mathbf {A} +\mathbf {B} &={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\\\end{bmatrix}}+{\begin{bmatrix}b_{11}&b_{12}&\cdots &b_{1n}\\b_{21}&b_{22}&\cdots &b_{2n}\\\vdots &\vdots &\ddots &\vdots \\b_{m1}&b_{m2}&\cdots &b_{mn}\\\end{bmatrix}}\\&={\begin{bmatrix}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}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}+b_{m1}&a_{m2}+b_{m2}&\cdots &a_{mn}+b_{mn}\\\end{bmatrix}}\\\end{aligned}},円\!}

Or more concisely (assuming that A + B = C):^{[4] [5]}

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

For example:

${\displaystyle {\begin{bmatrix}1&3\1円&0\1円&2\end{bmatrix}}+{\begin{bmatrix}0&0\7円&5\2円&1\end{bmatrix}}={\begin{bmatrix}1+0&3+0\1円+7&0+5\1円+2&2+1\end{bmatrix}}={\begin{bmatrix}1&3\8円&5\3円&3\end{bmatrix}}}${\displaystyle {\begin{bmatrix}1&3\1円&0\1円&2\end{bmatrix}}+{\begin{bmatrix}0&0\7円&5\2円&1\end{bmatrix}}={\begin{bmatrix}1+0&3+0\1円+7&0+5\1円+2&2+1\end{bmatrix}}={\begin{bmatrix}1&3\8円&5\3円&3\end{bmatrix}}}

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:

${\displaystyle {\begin{bmatrix}1&3\1円&0\1円&2\end{bmatrix}}-{\begin{bmatrix}0&0\7円&5\2円&1\end{bmatrix}}={\begin{bmatrix}1-0&3-0\1円-7&0-5\1円-2&2-1\end{bmatrix}}={\begin{bmatrix}1&3\\-6&-5\\-1&1\end{bmatrix}}}${\displaystyle {\begin{bmatrix}1&3\1円&0\1円&2\end{bmatrix}}-{\begin{bmatrix}0&0\7円&5\2円&1\end{bmatrix}}={\begin{bmatrix}1-0&3-0\1円-7&0-5\1円-2&2-1\end{bmatrix}}={\begin{bmatrix}1&3\\-6&-5\\-1&1\end{bmatrix}}}

• 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.

• Matrix Algebra and R

• Linear algebra
• Bilinear maps

• Use American English from January 2019
• All Wikipedia articles written in American English
• Articles with short description
• Short description matches Wikidata