Lecture 3 - What is "echelon form"?

Learning Objectives

• Understand the definitions of echelon form and reduced echelon form.
• Given a matrix, identify whether the matrix is in echelon form, reduced echelon form, or neither.

Echelon Form

The word "echelon" is a military term that describes a formation in which each unit is to the left or right of the unit in front of it. For example, the planes below are flying in an echelon formation:

In this course, "echelon form" will describe the position of entries in a matrix. In order for us to understand what it means for a matrix to be in echelon form, we will need to consider the leading entries of each row of a matrix.

Definition. In a matrix, the leading entry of a row is the first non-zero entry in the row, reading from left to right. If a row consists of all zeroes, then that row does not have a leading entry.

Example 1. Consider this matrix: \begin{bmatrix} 0 & 0 & 0 & 0 & 8 \\ -1 & 2 & 5 & -7 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 3 & 0 & 6 \end{bmatrix} The leading entry of the first row is 8, in column 5 of that row. The leading entry of the second row is -1, in the first column. The third row of this matrix does not have a leading entry, since it is all zeroes. Finally, the leading entry of the fourth row is 3, in the third column. \( \Box \)

Definition. A matrix is in echelon form if it satisfies all three of these conditions:

1. Any rows of all zeroes are below any other rows.
2. Each leading entry of a row (other than the first row) is to the right of the leading entry in the row above it.
3. All entries in a column below a leading entry are zeroes.

Example 1 revisited. Is this matrix in echelon form? \begin{bmatrix} 0 & 0 & 0 & 0 & 8 \\ -1 & 2 & 5 & -7 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 3 & 0 & 6 \end{bmatrix} Condition 1 is not satisfied by this matrix since the third row is all zeroes but it is not below all other rows. Condition 2 is not satisfied since the leading entry of Row 2 is to the left of the leading entry in Row 1. Condition 3 is not satisfied because the leading entry in the first row has non-zero entries below it. This matrix is not in echelon form. \( \Box \)

Note that in the previous example, none of the conditions for echelon form were satisfied. If any condition is false for a matrix, even once, then that matrix is not in echelon form. All of the conditions must be true for the matrix to be in echelon form.

Example 2. Is this matrix in echelon form? \begin{bmatrix} 3 & 2 & 0 & 7 & 9 \\ 0 & 4 & 5 & 10 & 0 \\ 0 & 0 & 0 & -4 & 1 \\ 0 & 0 & 0 & 0 & 6 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} If we check all three conditions, we see that this matrix is in echelon form. \( \Box \)

You might have noticed that if Conditions 1 and 2 for echelon form are satisfied, then Condition 3 must automatically be satisfied. We could leave Condition 3 out of our definition, but we will see that this condition is useful to think about when considering row operations for putting a matrix into echelon form.

In the next few lectures, we will learn how echelon form allows us to easily determine the nature of the solutions of a corresponding system of linear equations. If we want to actually describe the solutions themselves, it is more useful to consider a matrix in "reduced" echelon form.

Reduced Echelon Form

Definition. A matrix is in reduced echelon form if it satisfies all three of these conditions:

1. Any rows of all zeroes are below any other rows.
2. Each leading entry of a row (other than the first row) is to the right of the leading entry in the row above it.
3. All entries in a column below a leading entry are zeroes.
4. The leading entry of each row is 1.
5. Each leading entry is the only non-zero entry in its column.

Note that Conditions 1-3 of this definition are the same as the definition of echelon form.

Example 3. Is this matrix in reduced echelon form? \begin{bmatrix} 1 & 0 & 3 & 0 & 0 & 9 \\ 0 & 1 & 4 & 0 & 0 & -6 \\ 0 & 0 & 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} If we check all five conditions, we see that this matrix is in reduced echelon form. \( \Box \)

We already discussed how Condition 3 is automatically satisfied if Conditions 1 and 2 are. We also see that Condition 5 is stricter than Condition 3, since it requires zeroes to be above and below any leading entries. It is not true that Condition 5 is automatically satisfied when the other conditions are, so for reduced echelon form, we need to check that condition directly.

As we will see in the next lecture, we can use row operations (which we learned about in Lecture 2) to transform any given matrix into a corresponding echelon form. There may be many different echelon forms corresponding to a matrix. However, there is only one unique reduced echelon form for any given matrix. Proof of this uniqueness is beyond the scope of this course, but you can find a proof here.