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

##### 1 Answer

**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

In row echelon form:

#((color(blue)(1),2,3,4),(0,0,color(blue)(1),3),(0,0,0,color(blue)(1)),(color(blue)(0),color(blue)(0),color(blue)(0),color(blue)(0)))# #((color(blue)(1),2,3,4),(0,color(blue)(1),0,3),(0,0,color(blue)(1),color(blue)(1)),(color(blue)(0),color(blue)(0),color(blue)(0),color(blue)(0)))#

And in reduced echelon form:

#((color(blue)(1),2,0,0),(0,0,color(blue)(1),0),(0,0,0,color(blue)(1)),(color(blue)(0),color(blue)(0),color(blue)(0),color(blue)(0)))# #((color(blue)(1),0,0,0),(0,color(blue)(1),0,0),(0,0,color(blue)(1),color(blue)(1)),(color(blue)(0),color(blue)(0),color(blue)(0),color(blue)(0)))#

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

#((color(blue)(1),2,3,4),(color(red)(1),0,color(red)(2),3),(color(red)(0),color(red)(0),color(red)(0),color(red)(0)),(0,0,0,color(blue)(1)))# #((color(blue)(1),2,3,4),(0,color(blue)(1),0,3),(color(red)(0),color(red)(0),color(red)(0),color(red)(0)),(0,color(red)(1),color(blue)(1),color(blue)(1)))#

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