Matrix Addition

Definition: Matrix Addition and Subtraction

The addition and subtraction of two $m\times n$-matrices $A,B\in F^{m\times n}$ over the same field $F$ is done component-wise through the addition operation of $F$ and also results in an $m\times n$-matrix:

$$\left[\begin{array}{ccc}{a}_{11}& \cdots & a_{1n}\\ ⋮& \ddots & ⋮\\ a_{m1}& \cdots & a_{mn}\end{array}\right]\pm \left[\begin{array}{ccc}{b}_{11}& \cdots & b_{1n}\\ ⋮& \ddots & ⋮\\ b_{m1}& \cdots & b_{mn}\end{array}\right]\stackrel{\text{def}}{=}\left[\begin{array}{ccc}{a}_{11}\pm b_{11}& \cdots & a_{1n}\pm b_{1n}\\ ⋮& \ddots & ⋮\\ a_{m1}\pm b_{m1}& \cdots & a_{mn}\pm b_{mn}\end{array}\right]\in F^{m\times n}$$