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Reduced Row Echelon Form(REF)



Row Echelon Form
Echelon Form Notation
Reduced Row Echelon Form
Uniqueness of RREF
Pivot Columns and Free Columns
Reading Solutions from REF
Reading Solutions from RREF
Parametric Vector Form
Detecting Inconsistency
Echelon Form of Special Matrices
Echelon Form and Rank
Summary: Everything Readable from RREF
Echelon Form FAQ



The Staircase That Reveals Everything

Row echelon form and reduced row echelon form are the standard targets of Gaussian elimination. The staircase pattern of pivots immediately exposes the rank, the free variables, and the solvability of the system. RREF goes one step further — it is unique for every matrix, making it a canonical form from which the solution can be read without any back substitution.



Row Echelon Form

A matrix is in row echelon form (REF) if it satisfies three conditions. Every zero row (a row of all zeros) sits at the bottom of the matrix. The leading entry of each nonzero row — called the pivot — is strictly to the right of the pivot in the row above. Every entry below a pivot is zero.

Pivot Definition
pivot=leading nonzero entry of a row in echelon form\text{pivot} = \text{leading nonzero entry of a row in echelon form}

These conditions create a staircase pattern that descends from upper-left to lower-right:

(23−147015020004−300000)\begin{pmatrix} \boxed{2} & 3 & -1 & 4 & 7 \\ 0 & \boxed{1} & 5 & 0 & 2 \\ 0 & 0 & 0 & \boxed{4} & -3 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}


The boxed entries are pivots. Each sits to the right of and below the previous one, with zeros filling in below and to the left. The bottom row is all zeros. In general schematic form, with ∗\boxed{\ast} marking pivots and ∙\bullet any entry:

Row Echelon Form Definition
REF: (∗∙∙∙∙0∗∙∙∙000∗∙00000)\text{REF: } \begin{pmatrix} \boxed{\ast} & \bullet & \bullet & \bullet & \bullet \\ 0 & \boxed{\ast} & \bullet & \bullet & \bullet \\ 0 & 0 & 0 & \boxed{\ast} & \bullet \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}

REF is not unique. Different sequences of row operations applied to the same matrix can produce different echelon forms — the pivots may have different values, and the non-pivot entries above the pivots may differ. What remains the same across all echelon forms of a given matrix is the set of pivot positions (which columns contain pivots).
120.504.50010.5020011.83
Echelon form: a staircase of pivots

Each pivot sits strictly to the right of the one above it and every entry below a pivot is zero, which is what makes the shape a staircase. Rows of all zeros, if any, have sunk to the bottom. The form reached depends on the route taken through the elimination. Try a different pivot order and compare on the Gaussian elimination visualizer.

This is as far as plain Gaussian elimination goes; everything below this section is about what the extra reduction buys.

Echelon Form Notation

Notation

Echelon Form Notation

Two acronyms that differ by one word and by whether the article in front of them may be "the", a marked entry that is a position rather than a value, and a solution written with letters that stand for nothing in particular. All of them are catalogued among the linear algebra symbols.
The augmented bar [A∣b][A \mid \mathbf{b}], the row-operation arrows and A∼BA \sim B row equivalence come from Gaussian elimination notation; the m×nm \times n shape and entry brackets from matrix operation notation; rank⁡\operatorname{rank} from rank notation.
REF, RREF
row echelon form; reduced row echelon form
Two acronyms one word apart, and the extra word changes the grammar around them. REF is not unique — as this section notes, different row operations give different echelon forms — so a matrix has *an* REF. RREF is unique, which is what licenses the definite article: *the* RREF of a matrix, the subject of Uniqueness of RREF below.
Also writtenSpelled out in full in most European texts, which avoids the acronym entirely; some sources write rref⁡(A)\operatorname{rref}(A) as an operator, treating the unique reduced form as a function of the matrix — a spelling only the reduced version can support, precisely because it is unique.
Do not confuseUsing "the" with REF. Saying *the* echelon form of a matrix quietly asserts a uniqueness that only the reduced form has; what every echelon form does share is the set of pivot positions, not the entries.
2\boxed{2}, ∗\boxed{\ast}
a boxed entry marks a pivot — the leading nonzero entry of its row
The box is presentational, not algebraic: it highlights which entries are pivots without changing them, and the schematic form replaces the values with ∗\ast to show only the staircase. Circling is the common alternative on a blackboard. A pivot position is a location in the array; a pivot column is a column containing one.
CasesIn RREF every pivot is forced to 11 — the "leading 1" — so the box becomes redundant and texts drop it; in plain REF the pivot can be any nonzero value, which is exactly why it needs marking there and not here.
Do not confuseA pivot's value with its position. What is invariant across all echelon forms of a matrix is where the pivots sit, not what they are — so "the pivots of AA" means the positions, and it is the positions that determine rank.
x=p+tv1+sv2\mathbf{x} = \mathbf{p} + t\mathbf{v}_1 + s\mathbf{v}_2
the general solution: a particular vector plus free directions
Each free column contributes one letter, and the letters are arbitrary: tt, ss, or t1,t2,…t_1, t_2, \ldots when there are many — Parametric Vector Form below assembles them. A free variable is one whose column holds no pivot, so the pivot pattern decides how many letters the answer needs.
CasesThe count is forced, not chosen: free variables =n−rank⁡(A)= n - \operatorname{rank}(A), so the number of parameters is fixed even though their names are not. A unique solution is the case where that count is zero and the parameter part disappears entirely.
Do not confuseReading the parameters as unknowns still to be solved for. They are not — they range over all scalars, and the expression is the whole solution set at once rather than an equation waiting to be finished.

Reduced Row Echelon Form

A matrix is in reduced row echelon form (RREF) if it satisfies two additional conditions beyond REF. Each pivot is equal to 11. Each pivot is the only nonzero entry in its column — all entries above the pivot are also zero, not just below.

Applying these conditions to the previous example:

(10−160294015020001−3400000)\begin{pmatrix} 1 & 0 & -16 & 0 & \frac{29}{4} \\ 0 & 1 & 5 & 0 & 2 \\ 0 & 0 & 0 & 1 & -\frac{3}{4} \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}


Every pivot is 11, and every other entry in a pivot column is 00. The pivot columns are clean unit vectors within the matrix. The non-pivot columns (columns 33 and 55 in this example) can contain anything. In general schematic form:

Reduced Row Echelon Form Definition
RREF: (10∙0∙01∙0∙0001∙00000)\text{RREF: } \begin{pmatrix} \boxed{1} & 0 & \bullet & 0 & \bullet \\ 0 & \boxed{1} & \bullet & 0 & \bullet \\ 0 & 0 & 0 & \boxed{1} & \bullet \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}

RREF is achieved from REF by Gauss-Jordan elimination: scale each pivot row so the pivot becomes 11, then use row addition to eliminate all entries above each pivot.
1001.420101.080011.83
Reduced form: pivots alone in their columns

Every pivot is one and is the only non-zero entry in its column. Unlike the echelon form above, this one does not depend on how the elimination was carried out — every route ends at the same matrix, which is what makes uniqueness claims about it possible. Verify that by taking a different route on the Gaussian elimination visualizer.

Uniqueness is what lets the reduced form be used as a canonical description of the matrix rather than merely a convenient one.

Uniqueness of RREF

Every matrix has exactly one RREF. No matter which sequence of row operations is used to reach it, the result is the same:

RREF Uniqueness
RREF(A) is unique\text{RREF}(A) \text{ is unique}

This is not true of REF — different reduction paths can produce different row echelon forms of the same matrix, with different values in the non-pivot entries. But RREF is canonical: it is the unique representative of the matrix's row-equivalence class in reduced form.

The uniqueness of RREF implies that the pivot positions are intrinsic to the matrix. They do not depend on how the reduction is carried out. This means the rank (the number of pivots), the free variables (the non-pivot columns), and the entire solution structure are determined by the matrix itself, not by any particular algorithm applied to it.
Aspect Row Echelon Form (REF) Reduced Row Echelon Form (RREF)
Defining conditions zero rows at the bottom; pivots staircase strictly right; entries below each pivot are 0 all REF conditions + each pivot equals 1 + each pivot is the only nonzero entry in its column
Pivot values any nonzero number always 1
Entries above pivots anything (not constrained) always 0
Uniqueness not unique — different row-operation paths give different REFs (pivot positions stay the same) unique — every matrix has exactly one RREF, regardless of reduction path
Reached by forward elimination only Gauss-Jordan (forward + backward, plus pivot scaling)
Reading the solution by back substitution from the bottom row upward by direct inspection — each pivot variable is already isolated

Pivot Columns and Free Columns

Pivot columns are the columns that contain a pivot position in the echelon form. Free columns are everything else.

The number of pivot columns equals the rank of the matrix. The number of free columns in the coefficient matrix AA (not counting the augmented column) is the nullity — the dimension of the null space:

Free Variables Count
(number of free variables)=n−rank(A)\text{(number of free variables)} = n - \text{rank}(A)

In the context of a linear system, each pivot column corresponds to a pivot variable: a variable whose value is determined once the free variables are assigned. Each free column corresponds to a free variable: a parameter that can take any real value, generating a family of solutions.

For the RREF example above, columns 11, 22, and 44 are pivot columns, while columns 33 and 55 are free. The variables x1x_1, x2x_2, x4x_4 are pivot variables determined by x3x_3 and x5x_5, which are free.

Reading Solutions from REF

In row echelon form, the solution is extracted by back substitution: solve from the bottom nonzero row upward, one pivot variable at a time.

Worked Example


(13−25014−10012)\left(\begin{array}{ccc|c} 1 & 3 & -2 & 5 \\ 0 & 1 & 4 & -1 \\ 0 & 0 & 1 & 2 \end{array}\right)


Row 33: x3=2x_3 = 2.

Row 22: x2+4(2)=−1x_2 + 4(2) = -1, so x2=−9x_2 = -9.

Row 11: x1+3(−9)−2(2)=5x_1 + 3(-9) - 2(2) = 5, so x1=5+27+4=36x_1 = 5 + 27 + 4 = 36.

Each step uses values computed in previous steps. The process requires nn substitutions for an n×nn \times n system with a unique solution. When free variables are present, they appear as unresolved symbols carried through the substitution, producing a parametric expression.

Reading Solutions from RREF

In RREF, each pivot variable is already isolated. The solution is visible by inspection — no back substitution is needed.

Worked Example


The same system in RREF:

(10036010−90012)\left(\begin{array}{ccc|c} 1 & 0 & 0 & 36 \\ 0 & 1 & 0 & -9 \\ 0 & 0 & 1 & 2 \end{array}\right)


Reading directly: x1=36x_1 = 36, x2=−9x_2 = -9, x3=2x_3 = 2.

When free variables are present, RREF isolates each pivot variable in terms of the free variables:

(1030701−20100014)\left(\begin{array}{cccc|c} 1 & 0 & 3 & 0 & 7 \\ 0 & 1 & -2 & 0 & 1 \\ 0 & 0 & 0 & 1 & 4 \end{array}\right)


Column 33 is free. Reading off: x1=7−3x3x_1 = 7 - 3x_3, x2=1+2x3x_2 = 1 + 2x_3, x4=4x_4 = 4, and x3x_3 is free. Setting x3=tx_3 = t gives the parametric solution without any backward solving.

Parametric Vector Form

When free variables exist, the general solution is best expressed in parametric vector form. Each free variable becomes a parameter, and the solution is written as a particular solution plus a linear combination of direction vectors — one per free variable.

For the system with RREF

(120−150013−2)\left(\begin{array}{cccc|c} 1 & 2 & 0 & -1 & 5 \\ 0 & 0 & 1 & 3 & -2 \end{array}\right)


Pivots in columns 11 and 33. Free variables: x2=sx_2 = s, x4=tx_4 = t. Reading off: x1=5−2s+tx_1 = 5 - 2s + t, x3=−2−3tx_3 = -2 - 3t.

x=(50−20)+s(−2100)+t(10−31)\mathbf{x} = \begin{pmatrix} 5 \\ 0 \\ -2 \\ 0 \end{pmatrix} + s\begin{pmatrix} -2 \\ 1 \\ 0 \\ 0 \end{pmatrix} + t\begin{pmatrix} 1 \\ 0 \\ -3 \\ 1 \end{pmatrix}


The first vector is a particular solution xp\mathbf{x}_p. The two direction vectors form a basis for the null space of the coefficient matrix. The solution set is the null space translated by xp\mathbf{x}_p — an affine subspace of dimension 22 in R4\mathbb{R}^4.

For a homogeneous system (b=0\mathbf{b} = \mathbf{0}), the particular solution is 0\mathbf{0} and the general solution is purely a null-space combination.

Detecting Inconsistency

A system is inconsistent — has no solution — if and only if the echelon form of the augmented matrix contains a row of the form

[00⋯0∣d]with d≠0[0 \quad 0 \quad \cdots \quad 0 \mid d] \qquad \text{with } d \neq 0


This row represents the equation 0=d0 = d, which no values of the unknowns can satisfy.

In terms of rank, inconsistency occurs when rank([A∣b])>rank(A)\text{rank}([A \mid \mathbf{b}]) > \text{rank}(A). The extra column b\mathbf{b} introduces a new pivot that the coefficient matrix alone does not have — meaning b\mathbf{b} is not in the column space of AA.

If no such contradictory row appears, the system is consistent. The distinction between unique and infinite solutions then depends on whether free variables exist: rank =n= n gives a unique solution, rank <n< n gives infinitely many.
000000000000
The contradiction row

All coefficients gone, constant still present: the row claims zero equals a non-zero number. Detecting inconsistency needs no separate test, because reduction surfaces this row on its own if the system has no solution. Force one to appear on the Gaussian elimination visualizer.

Spotting this row early saves the rest of the reduction, since no later step can repair it.

Echelon Form of Special Matrices

Several matrix types are already in echelon form or reach it with minimal work.

The identity matrix InI_n is in RREF: nn pivots, each equal to 11, each alone in its column. Rank nn, no free variables.

Any diagonal matrix is in REF. If all diagonal entries are nonzero, scaling each to 11 produces RREF. If some diagonal entries are zero, the corresponding rows are zero rows and the rank drops.

An upper triangular matrix is already in REF. Its pivots are the nonzero diagonal entries, and its rank equals the number of nonzero entries on the diagonal.

A lower triangular matrix is not in REF (the staircase goes the wrong way), but forward elimination converts it quickly — each column requires at most one elimination step.

The zero matrix is in both REF and RREF. It has rank 00, and the corresponding homogeneous system Ox=0O\mathbf{x} = \mathbf{0} has every variable free.
Matrix type Echelon status Pivot structure Rank
Identity In already in RREF n pivots, each equal to 1, each alone in its column n
Diagonal — all entries nonzero in REF; reaches RREF after scaling each row one pivot per row, equal to the diagonal entry n
Diagonal — some entries zero in REF (zero rows reorder to the bottom) a pivot only where the diagonal entry is nonzero number of nonzero diagonal entries
Upper triangular already in REF nonzero diagonal entries serve as pivots number of nonzero diagonal entries
Lower triangular NOT in REF — the staircase goes the wrong direction forward elimination converts it quickly; at most one elimination per column depends on the matrix
Zero matrix O already in both REF and RREF no pivots; every column is free 0

Echelon Form and Rank

The rank of any matrix equals the number of pivots in any echelon form. This is the standard computational method for determining rank: row reduce and count pivots.

From the rank, everything else follows. The dimension of the column space is rr. The dimension of the row space is rr. The number of free variables is n−rn - r. The dimension of the null space is n−rn - r. The system Ax=bA\mathbf{x} = \mathbf{b} is consistent if and only if the augmented matrix has the same rank rr as AA. The solution, when it exists, is unique if r=nr = n and infinite if r<nr < n.

The pivot positions also identify bases for the fundamental subspaces. The original columns at pivot positions form a basis for the column space. The nonzero rows of the echelon form are a basis for the row space. The parametric solution of Ax=0A\mathbf{x} = \mathbf{0} gives a basis for the null space. Everything flows from counting and locating the pivots.

Summary: Everything Readable from RREF

The RREF of A is a single canonical form, but it carries a remarkable amount of structural information about the matrix — the rank, the nullity, bases for the column, row, and null spaces, and the consistency, uniqueness, and invertibility verdicts for any system A·x = b. The table below collects each property that can be extracted directly from RREF alongside the rule for reading it and the underlying reason.
The readings below are grouped by where in the reduced form each answer sits, because that is what makes them one computation rather than eight. Two come from counting pivots, three from noting which positions hold them, and three from the augmented column. Nothing needs a second reduction, and nothing needs the original matrix again except where the entry says so explicitly.
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Everything readable from the RREF

One reduction answers all of these. What differs is where in the reduced form the answer sits — the pivot positions, the free columns, or the augmented column.

8readings
From the pivot count2
1
Rank§ 10
count them; nothing else needed
rank⁡(A)=\operatorname{rank}(A) = number of pivots
Pivots correspond to independent rows and to independent columns simultaneously, which is why row rank equals column rank — they are the same pivots counted two ways.
2
Nullity§ 10
the number of free columns
n−rank⁡(A)n - \operatorname{rank}(A)
Every column without a pivot is a free variable, and every free variable contributes one dimension to the null space. Rank-nullity is not a separate theorem here so much as a restatement of the column count.
From the pivot positions — bases3
3
Column space basis§ 4
not of the reduced form
pivot columns of the original AA
Row operations change the column space but preserve which columns are dependent on which. So the reduction tells you which columns to take, and you take them from AA — the single most common error on this page.
4
Row space basis§ 4
of the reduced form, not the original
nonzero rows of the RREF
The exact opposite of the rule above, and the reason both are stated here together. Row operations preserve the row space itself, so the reduced rows still span it — and they are independent, which the original rows may not be.
5
Null space basis§ 7
set each free variable to 11 in turn
one vector per free variable
Read from the parametric vector form. The count matches the nullity above, and each basis vector is the solution obtained by turning one free variable on and the rest off.
From the augmented column3
6
Consistency§ 8
rank⁡(A)=rank⁡([A∣b])\operatorname{rank}(A) = \operatorname{rank}([A \mid \mathbf{b}])
no row [ 0⋯0∣d ][\,0 \cdots 0 \mid d\,], d≠0d \neq 0
Such a row asserts 0=d0 = d, so the system has no solution. Equivalently b\mathbf{b} lies outside the column space — the augmented column raises the rank, and that is what the contradiction row records.
7
Uniqueness§ 6
no free variables
a pivot in every column of AA
Consistency gives at least one solution, no free variables gives at most one. Both conditions are needed — an inconsistent system has none however many pivots it has.
8
Invertibility§ 9
square and full rank
rref⁡(A)=In\operatorname{rref}(A) = I_n
The two conditions above at once, for a square matrix: a pivot in every row and every column leaves the identity. This is one line of the invertibility equivalence, and reduction is how it is checked in practice.
Nothing below requires a second computation. That is the return on reducing: rank, nullity, three bases, consistency, uniqueness and invertibility all come out of a single echelon form, which is why Gaussian elimination is the one algorithm worth doing by hand.
Everything readable from the RREF·/linear-algebra/linear-systems/echelon-formLearn Math Class
The two basis rules are worth reading against each other. For the column space the reduction tells you which columns to take and you take them from the original matrix; for the row space you take the reduced rows themselves. They look symmetric and are not, because row operations preserve the row space while changing the column space — the pivots survive both, which is why one reduction serves both purposes.
The last three entries are the same question asked with increasing strictness: does a solution exist, is it unique, and does the matrix invert. Each adds a condition to the one before, and a square matrix satisfying all three reduces to the identity — which is the point at which reduction stops being a technique for solving systems and becomes a test of the matrix itself.

Echelon Form FAQ

Can you speak of "the" echelon form of a matrix?

+
Only for the reduced form, which is unique. Plain row echelon form is not: different elimination paths produce different entries, so the definite article quietly asserts a uniqueness that only RREF has. What every echelon form of a matrix does share is the set of pivot positions, and those are what determine the rank.Read more →

Does a pivot have to equal 11?

+
In reduced row echelon form yes, where each pivot is forced to 11 and earns the name leading one. In plain echelon form a pivot can be any nonzero value, which is exactly why it needs marking there and not in RREF. Note that the phrase "the pivots of AA" refers to positions rather than values.Read more →

Are the free parameters unknowns still waiting to be solved for?

+
No, they range over all scalars, and the expression containing them is the entire solution set written at once rather than an equation left unfinished. Their number is forced even though their names are not: there are n−rank⁡(A)n - \operatorname{rank}(A) of them, and a unique solution is simply the case where that count reaches zero.Read more →