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{align} \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{align},! Or more concisely (assuming that ): :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 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{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

References

References

1. Elementary Linear Algebra by Rorres Anton 10e p53
2. Weisstein, Eric W.. "Matrix Addition".
3. "Finding the Sum and Difference of Two Matrices {{!}} College Algebra".