Matrix addition

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

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

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

Definition

Two matrices must have an equal number of rows and columns to be added. 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:

\begin{matrix} 𝐀 + 𝐁 & = [\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{matrix}] + [\begin{matrix} b_{11} & b_{12} & \dots & b_{1 n} \\ b_{21} & b_{22} & \dots & b_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ b_{m 1} & b_{m 2} & \dots & b_{m n} \end{matrix}] \\ = [\begin{matrix} a_{11} + b_{11} & a_{12} + b_{12} & \dots & a_{1 n} + b_{1 n} \\ a_{21} + b_{21} & a_{22} + b_{22} & \dots & a_{2 n} + b_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} + b_{m 1} & a_{m 2} + b_{m 2} & \dots & a_{m n} + b_{m n} \end{matrix}] \end{matrix}

Or more concisely (assuming that A + B = C):

c_{i j} = a_{i j} + b_{i j}

For example:

[\begin{matrix} 1 & 3 \\ 1 & 0 \\ 1 & 2 \end{matrix}] + [\begin{matrix} 0 & 0 \\ 7 & 5 \\ 2 & 1 \end{matrix}] = [\begin{matrix} 1 + 0 & 3 + 0 \\ 1 + 7 & 0 + 5 \\ 1 + 2 & 2 + 1 \end{matrix}] = [\begin{matrix} 1 & 3 \\ 8 & 5 \\ 3 & 3 \end{matrix}]

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:

[\begin{matrix} 1 & 3 \\ 1 & 0 \\ 1 & 2 \end{matrix}] - [\begin{matrix} 0 & 0 \\ 7 & 5 \\ 2 & 1 \end{matrix}] = [\begin{matrix} 1 - 0 & 3 - 0 \\ 1 - 7 & 0 - 5 \\ 1 - 2 & 2 - 1 \end{matrix}] = [\begin{matrix} 1 & 3 \\ - 6 & - 5 \\ - 1 & 1 \end{matrix}]

Notes

↑ 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

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.

[1] Elementary Linear Algebra by Rorres Anton 10e p53

[FOOTNOTELipschutzLipson2017-2] Lipschutz & Lipson 2017.

[FOOTNOTERileyHobsonBence2006-3] Riley, Hobson & Bence 2006.

[4] Weisstein, Eric W. "Matrix Addition". mathworld.wolfram.com. Retrieved 2020-09-07.

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Matrix addition

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