Matrix addition
In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. For a vector, v → {\displaystyle {\vec {v}}\!} , adding two matrices would have the geometric effect of applying each matrix transformation separately onto v → {\displaystyle {\vec {v}}\!} , then adding the transformed vectors. A v → + B v → = ( A + B ) v → {\displaystyle \mathbf {A} {\vec {v}}+\mathbf {B} {\vec {v}}=(\mathbf {A} +\mathbf {B} ){\vec {v}}\!}
In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together.
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{\displaystyle \mathbf {A} {\vec {v}}+\mathbf {B} {\vec {v}}=(\mathbf {A} +\mathbf {B} ){\vec {v}}\!}
Definition
[edit]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]
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{\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}}\,\!}
Or more concisely (assuming that A + B = C):[4]
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{\displaystyle c_{ij}=a_{ij}+b_{ij}}
For example:
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{\displaystyle {\begin{bmatrix}1&3\\1&0\\1&2\end{bmatrix}}+{\begin{bmatrix}0&0\\7&5\\2&1\end{bmatrix}}={\begin{bmatrix}1+0&3+0\\1+7&0+5\\1+2&2+1\end{bmatrix}}={\begin{bmatrix}1&3\\8&5\\3&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:
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{\displaystyle {\begin{bmatrix}1&3\\1&0\\1&2\end{bmatrix}}-{\begin{bmatrix}0&0\\7&5\\2&1\end{bmatrix}}={\begin{bmatrix}1-0&3-0\\1-7&0-5\\1-2&2-1\end{bmatrix}}={\begin{bmatrix}1&3\\-6&-5\\-1&1\end{bmatrix}}}
See also
[edit]Notes
[edit]- ^ Elementary Linear Algebra by Rorres Anton 10e p53
- ^ Lipschutz & Lipson 2017.
- ^ Riley, Hobson & Bence 2006.
- ^ Weisstein, Eric W. "Matrix Addition". mathworld.wolfram.com. Retrieved 2020-09-07.
References
[edit]- 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.