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Matrix addition

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From Wikipedia, the free encyclopedia

Illustration of the addition of two matrices.

Notions of sums for matrices in linear algebra

Inmathematics,matrix addition is the operation of adding twomatrices by adding the corresponding entries together.

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

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

Definition

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

{\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 thatA +B =C):^[4]^[5]

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

For example:

{\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 ofA andB, denotedA −B, is computed by subtracting elements ofB from corresponding elements ofA, and has the same dimensions asA andB. For example:

{\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}}

Notes

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^Elementary Linear Algebra by Rorres Anton 10e p53
^Lipschutz & Lipson 2017.
^Riley, Hobson & Bence 2006.
^Weisstein, Eric W."Matrix Addition".mathworld.wolfram.com. Retrieved2020-09-07.
^"Finding the Sum and Difference of Two Matrices | College Algebra".courses.lumenlearning.com. Retrieved2020-09-07.

References

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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.