Row echelon form

Key takeaways

Row echelon form belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Row echelon form to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Row echelon form from memory before moving on to harder problems.

Reference excerpt

In linear algebra, a matrix is in row echelon form if it can be obtained as the result of Gaussian elimination. Every matrix can be put in row echelon form by applying a sequence of elementary row operations. The term echelon comes from the French échelon ("level" or step of a ladder), and refers to the fact that the nonzero entries of a matrix in row echelon form look like the steps of a staircase.

For square matrices, an upper triangular matrix with nonzero entries on the diagonal is in row echelon form, and a matrix in row echelon form is (weakly) upper triangular. Thus, the row echelon form can be viewed as a generalization of upper triangular form for rectangular matrices. A matrix is in reduced row echelon form if it is in row echelon form, with the additional property that the first nonzero entry of each row is equal to 1 {\displaystyle 1} and is the only nonzero entry of its column. The reduced row echelon form of a matrix is unique and does not depend on the sequence of elementary row operations that is used to obtain it. The specific type of Gaussian elimination that transforms a matrix to reduced row echelon form is sometimes called Gauss–Jordan elimination. A matrix is in column echelon form if its transpose is in row echelon form. Since all properties of column echelon forms can therefore immediately be deduced from the corresponding properties of row echelon forms, only row echelon forms are considered in the remainder of the article.

Row echelon form

A matrix is in row echelon form if

All rows having only zero entries are at the bottom. The leading entry (that is, the leftmost non-zero entry) of every non-zero row, called the pivot, is to the right of the leading entry of every row above. Some texts add the condition that the leading coefficient must be 1 while others require this only in reduced row echelon form. These two conditions imply that all entries in a column below a leading coefficient are zeros. The following is an example of a 4 × 5 {\displaystyle 4\times 5} matrix in row echelon form, but not in reduced row echelon form:

[ 1 a 0 a 1 a 2 a 3 0 0 2 a 4 a 5 0 0 0 1 a 6 0 0 0 0 0 ] {\displaystyle \left[{\begin{array}{ccccc}1&a_{0}&a_{1}&a_{2}&a_{3}\\0&0&2&a_{4}&a_{5}\\0&0&0&1&a_{6}\\0&0&0&0&0\end{array}}\right]}

Many properties of matrices may be easily deduced from their row echelon form, such as the rank and the kernel.

Reduced row echelon form A matrix is in reduced row echelon form (also called row canonical form) if it satisfies the following conditions:

It is in row echelon form. The leading entry in each nonzero row is 1 (called a leading one). Each column containing a leading 1 has zeros in all its other entries. If the first two conditions are verified, the last condition is equivalent to:

Each column containing a leading 1 has zeros in all entries above the leading 1. While a matrix may have several echelon forms, its reduced echelon form is unique. Given a matrix in reduced row echelon form, if one permutes the columns in order to have the leading 1 of the ith row in the ith column, one gets a matrix of the form

( I X 0 0 ) , {\displaystyle {\begin{pmatrix}I&X\\0&0\end{pmatrix}},}

… excerpt ends here. Continue reading the full article.

Illustrations

Row echelon form: Example of a rectangular matrix in row echelon form — Example of a rectangular matrix in row echelon form

Worked examples

Example 1 — a first encounter with Row echelon form

Start with the simplest possible case. Write down what Row echelon form claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Row echelon form before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Row echelon form ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Row echelon form

In research: Row echelon form appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Row echelon form in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Row echelon form is common in secondary-school and first-year university syllabi. It links to neighbouring topics Matrix normal forms, Numerical linear algebra, so understanding it makes those chapters shorter.
In everyday life: Look for Row echelon form outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Row echelon form in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Row echelon form means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Row echelon form out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Row echelon form in simple terms?

In linear algebra, a matrix is in row echelon form if it can be obtained as the result of Gaussian elimination. Every matrix can be put in row echelon form by applying a sequence of elementary row operations.

Why does Row echelon form matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Row echelon form?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Row echelon form.