# How to determine whether a matrices are in echelon form, reduced echelon form or not in echelon form ?

Nov 20, 2015

Row echelon form implies that:

• The leading (first) entry in each row must be $1$.
• The leading entry on each subsequent row must be on a new column to the right
• All rows where all entries are zero are below rows where NOT all entries are zero

Reduced echelon form further follows from echelon form conditions, provided that in each column, the leading entry is the only nonzero entry in its column.

I have some examples below where I've highlighted the entries in line with the conditions as $\textcolor{b l u e}{\text{blue}}$, and those breaking the conditions as $\textcolor{red}{\text{red}}$.

In row echelon form:

$\left(\begin{matrix}\textcolor{b l u e}{1} & 2 & 3 & 4 \\ 0 & 0 & \textcolor{b l u e}{1} & 3 \\ 0 & 0 & 0 & \textcolor{b l u e}{1} \\ \textcolor{b l u e}{0} & \textcolor{b l u e}{0} & \textcolor{b l u e}{0} & \textcolor{b l u e}{0}\end{matrix}\right)$ $\left(\begin{matrix}\textcolor{b l u e}{1} & 2 & 3 & 4 \\ 0 & \textcolor{b l u e}{1} & 0 & 3 \\ 0 & 0 & \textcolor{b l u e}{1} & \textcolor{b l u e}{1} \\ \textcolor{b l u e}{0} & \textcolor{b l u e}{0} & \textcolor{b l u e}{0} & \textcolor{b l u e}{0}\end{matrix}\right)$

And in reduced echelon form:

$\left(\begin{matrix}\textcolor{b l u e}{1} & 2 & 0 & 0 \\ 0 & 0 & \textcolor{b l u e}{1} & 0 \\ 0 & 0 & 0 & \textcolor{b l u e}{1} \\ \textcolor{b l u e}{0} & \textcolor{b l u e}{0} & \textcolor{b l u e}{0} & \textcolor{b l u e}{0}\end{matrix}\right)$ $\left(\begin{matrix}\textcolor{b l u e}{1} & 0 & 0 & 0 \\ 0 & \textcolor{b l u e}{1} & 0 & 0 \\ 0 & 0 & \textcolor{b l u e}{1} & \textcolor{b l u e}{1} \\ \textcolor{b l u e}{0} & \textcolor{b l u e}{0} & \textcolor{b l u e}{0} & \textcolor{b l u e}{0}\end{matrix}\right)$

If we were to construct matrices NOT in echelon form, we can modify the row echelon ones, like this:

$\left(\begin{matrix}\textcolor{b l u e}{1} & 2 & 3 & 4 \\ \textcolor{red}{1} & 0 & \textcolor{red}{2} & 3 \\ \textcolor{red}{0} & \textcolor{red}{0} & \textcolor{red}{0} & \textcolor{red}{0} \\ 0 & 0 & 0 & \textcolor{b l u e}{1}\end{matrix}\right)$ $\left(\begin{matrix}\textcolor{b l u e}{1} & 2 & 3 & 4 \\ 0 & \textcolor{b l u e}{1} & 0 & 3 \\ \textcolor{red}{0} & \textcolor{red}{0} & \textcolor{red}{0} & \textcolor{red}{0} \\ 0 & \textcolor{red}{1} & \textcolor{b l u e}{1} & \textcolor{b l u e}{1}\end{matrix}\right)$

(and of course, if row echelon conditions are not satisfied, reduced echelon conditions are also not fully satisfied.)