Understanding the definition of row echelon form from Golan.

Question

The definition is given below:

But I do not understand what is $s(i)$ and how to know it, could anyone give me a numerical example to explain the definition,please?

Answer 1

It means that for each row $i$, all elements before a certain column $j=s(i)$ are $0$, and that the function $i\longmapsto j=s(i)$ is (strictly) increasing, i.e. the number of $0$s at the beginning of a row is increasing.

This condition is not satisfied in the counterexample they give, as $s(1)=1, s(2)=3, s(3)=3, s(4)=5$.

Answer 2

$s(i)$ is the position (column number) of the first nonzero entry in row $i$. In the last matrix in your example $s(2) = s(3) = 3$.

If row $i$ is all $0$ then $s(i)=n+1$ even though there are only $n$ columns.