The element form shows the relationship between elementary transformation and elementary matrix

The relationship between elementary transformation of matrix and elementary matrix , Generally speaking, it is : Left row right column . That is to say, the left multiplication elementary matrix performs row elementary transformation ; And the right multiplication of elementary matrix is the column elementary transformation .

hypothesis
$$A_{m\times n}=\left(\begin{array}{cccc}{a}_{11}& a_{12}& ...& a_{1n}\\ ...& ...& ...& ...\\ a_{i1}& a_{i2}& ...& a_{in}\\ ...& ...& ...& ...\\ a_{j1}& a_{j2}& ...& a_{jn}\\ ...& ...& ...& ...\\ a_{m1}& a_{m2}& ...& a_{mn}\end{array}\right)$$
Then there are the following situations :

$$AE(i,j(c))=\left(\begin{array}{ccccccc}{a}_{11}& ...& a_{1i}& ...& a_{1j}+c{a}_{1i}& ...& a_{1n}\\ a_{21}& ...& a_{2i}& ...& a_{2j}+c{a}_{2i}& ...& a_{2n}\\ ...& ...& ...& ...& ...& ...& ...\\ a_{m1}& ...& a_{mi}& ...& a_{mj}+c{a}_{mi}& ...& a_{mn}\end{array}\right)$$