Matrix addition

1. Addition of 2 matrices

Matrix addition is the operation of adding two matrices by adding the corresponding entries together. Given two matrices $A$, $B$ that must have an equal number of rows and columns $(m\times n)$.

$$\begin{array}{rl}\left[\begin{array}{c}A\end{array}\right]+\left[\begin{array}{c}B\end{array}\right]& =\left[\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{21}& a_{22}& \dots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \dots & a_{mn}\end{array}\right]+\left[\begin{array}{cccc}{b}_{11}& b_{12}& \dots & b_{1n}\\ b_{21}& b_{22}& \dots & b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ b_{m1}& b_{m2}& \dots & b_{mn}\end{array}\right]\\ & =\left[\begin{array}{cccc}{a}_{11}+b_{11}& a_{12}+b_{12}& \dots & a_{1n}+b_{1n}\\ a_{21}+b_{21}& a_{22}+b_{22}& \dots & a_{2n}+b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}+b_{m1}& a_{m2}+b_{m2}& \dots & a_{mn}+b_{mn}\end{array}\right]\end{array}$$

$$A+B=C\Rightarrow c_{ij}=a_{ij}+b_{ij}$$

For example

$$\left[\begin{array}{cc}1& 2\\ 3& 4\\ 5& 6\end{array}\right]+\left[\begin{array}{cc}0& 6\\ 1& 2\\ 3& 0\end{array}\right]=\left[\begin{array}{cc}1& 8\\ 4& 6\\ 8& 6\end{array}\right]$$

2. Subtraction of 2 matrices

Similarly, we can also subtract one matrix from another, as long as they have the same dimensions.

$$\begin{array}{rl}\left[\begin{array}{c}A\end{array}\right]–\left[\begin{array}{c}B\end{array}\right]& =\left[\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{21}& a_{22}& \dots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \dots & a_{mn}\end{array}\right]–\left[\begin{array}{cccc}{b}_{11}& b_{12}& \dots & b_{1n}\\ b_{21}& b_{22}& \dots & b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ b_{m1}& b_{m2}& \dots & b_{mn}\end{array}\right]\\ & =\left[\begin{array}{cccc}{a}_{11}–b_{11}& a_{12}–b_{12}& \dots & a_{1n}–b_{1n}\\ a_{21}–b_{21}& a_{22}–b_{22}& \dots & a_{2n}–b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}–b_{m1}& a_{m2}–b_{m2}& \dots & a_{mn}–b_{mn}\end{array}\right]\end{array}$$

$$A–B=C\Rightarrow c_{ij}=a_{ij}–b_{ij}$$

For example

$$\left[\begin{array}{cc}1& 2\\ 3& 4\\ 5& 6\end{array}\right]–\left[\begin{array}{cc}0& 6\\ 1& 2\\ 3& 0\end{array}\right]=\left[\begin{array}{cc}1& -4\\ 2& 2\\ 2& 6\end{array}\right]$$