rref

Reduced row echelon form (Gauss-Jordan elimination)

Syntax

R = rref(A)

R = rref(A,tol)

[R,p] = rref(A)

Description

R = rref(A) returns the reduced row echelon form of A using Gauss-Jordan elimination with partial pivoting.

R = rref(A,tol) specifies a pivot tolerance that the algorithm uses to determine negligible columns.

[R,p] = rref(A) also returns the nonzero pivots p.

Examples

Reduced Row Echelon Form of Matrix

Create a matrix and calculate the reduced row echelon form. In this form, the matrix has leading 1s in the pivot position of each column.

A = magic(3)

A = 3×3

     8     1     6
     3     5     7
     4     9     2

RA = rref(A)

RA = 3×3

     1     0     0
     0     1     0
     0     0     1

The 3-by-3 magic square matrix is full rank, so the reduced row echelon form is an identity matrix.

Now, calculate the reduced row echelon form of the 4-by-4 magic square matrix. Specify two outputs to return the nonzero pivot columns. Since this matrix is rank deficient, the result is not an identity matrix.

B = magic(4)

B = 4×4

    16     2     3    13
     5    11    10     8
     9     7     6    12
     4    14    15     1

[RB,p] = rref(B)

RB = 4×4

     1     0     0     1
     0     1     0     3
     0     0     1    -3
     0     0     0     0

p = 1×3

     1     2     3

Row Reduction of Augmented Matrices

Use Gauss-Jordan elimination on augmented matrices to solve a linear system and calculate the matrix inverse. These techniques are mainly of academic interest, since there are more efficient and numerically stable ways to calculate these values.

Create a 3-by-3 magic square matrix. Add an additional column to the end of the matrix. This augmented matrix represents a linear system $\mathrm{Ax}=\mathit{b}$, with the extra column corresponding to $\mathit{b}$.

A = magic(3);
A(:,4) = [1; 1; 1]

A = 3×4

     8     1     6     1
     3     5     7     1
     4     9     2     1

Calculate the reduced row echelon form of A. Index into R to extract the entries in the extra (augmented) column, which contains the solution to the linear system.

R = rref(A)

R = 3×4

    1.0000         0         0    0.0667
         0    1.0000         0    0.0667
         0         0    1.0000    0.0667

x = R(:,end)

x = 3×1

    0.0667
    0.0667
    0.0667

A more efficient way to solve this linear system is with the backslash operator, x = A\b.

Create a similar magic square matrix, but this time append an identity matrix of the same size to the end columns.

A = [magic(3) eye(3)]

A = 3×6

     8     1     6     1     0     0
     3     5     7     0     1     0
     4     9     2     0     0     1

Calculate the reduced row echelon form of A. In this form the extra columns contain the inverse matrix for the 3-by-3 magic square matrix.

R = rref(A)

R = 3×6

    1.0000         0         0    0.1472   -0.1444    0.0639
         0    1.0000         0   -0.0611    0.0222    0.1056
         0         0    1.0000   -0.0194    0.1889   -0.1028

inv_A = R(:,4:end)

inv_A = 3×3

    0.1472   -0.1444    0.0639
   -0.0611    0.0222    0.1056
   -0.0194    0.1889   -0.1028

A more efficient way to calculate the inverse matrix is with inv(A).

Solve System of Equations

Consider a linear system of equations with four equations and three unknowns.

$$\begin{array}{l}{\mathit{x}}_1+{\mathit{x}}_2+5{\mathit{x}}_3=6\\ 2{\mathit{x}}_1+{\mathit{x}}_2+8{\mathit{x}}_3=8\\ {\mathit{x}}_1+2{\mathit{x}}_2+7{\mathit{x}}_3=10\\ -{\mathit{x}}_1+{\mathit{x}}_2-{\mathit{x}}_3=2.\end{array}$$

Create an augmented matrix that represents the system of equations.

A = [1  1  5;
     2  1  8;
     1  2  7;
    -1  1 -1];
b = [6 8 10 2]';
M = [A b];

Use rref to express the system in reduced row echelon form.

R = rref(M)

R = 4×4

     1     0     3     2
     0     1     2     4
     0     0     0     0
     0     0     0     0

The first two rows of R contain equations that express ${\mathit{x}}_1$ and ${\mathit{x}}_2$ in terms of ${\mathit{x}}_3$. The second two rows imply that there exists at least one solution that fits the right-hand side vector (otherwise one of the equations would read $1=0$). The third column does not contain a pivot, so ${\mathit{x}}_3$ is an independent variable. Therefore, there are infinitely many solutions for ${\mathit{x}}_1$ and ${\mathit{x}}_2$, and ${\mathit{x}}_3$ can be chosen freely.

$$\begin{array}{l}{\mathit{x}}_1=2-3{\mathit{x}}_3\\ {\mathit{x}}_2=4-2{\mathit{x}}_3.\end{array}$$

For example, if ${\mathit{x}}_3=1$, then ${\mathit{x}}_1=-1$ and ${\mathit{x}}_2=2$.

From a numerical standpoint, a more efficient way to solve this system of equations is with x0 = A\b, which (for a rectangular matrix A) calculates the least-squares solution. In that case, you can check the accuracy of the solution with norm(A*x0-b)/norm(b) and the uniqueness of the solution by checking if rank(A) is equal to the number of unknowns. If more than one solution exists, then they all have the form of $\mathit{x}={\mathit{x}}_0+\mathrm{nt}$, where $\mathit{n}$ is the null space null(A) and $\mathit{t}$ can be chosen freely.

Input Arguments

A — Input matrix

matrix

Input matrix.

Data Types: single | double
Complex Number Support: Yes

tol — Pivot tolerance

max(size(A))*eps*norm(A,inf) (default) | scalar

Pivot tolerance, specified as a scalar. If the largest element (by absolute value) in a pivot column is below the tolerance, then the column is zeroed out. This prevents division and multiplication with nonzero pivot elements smaller than the tolerance.

Data Types: single | double

Output Arguments

R — Reduced row echelon form of A

matrix

Reduced row echelon form of A, returned as a matrix.

p — Nonzero pivot columns

vector

Nonzero pivot columns, returned as a vector. Each element in p is a column index of A. You can use p to estimate several quantities:

• length(p) is an estimate of the rank of A.

• x(p) contains the pivot variables in a linear system Ax = b.

• A(:,p) is a basis for the range of A.

• R(1:r,p) is the r-by-r identity matrix, where r = length(p).

Limitations

• rank, orth, and null are typically faster and more accurate for computing the rank and basis vectors of a matrix.

• mldivide is recommended to solve linear systems.

More About

Partial Pivoting

Partial pivoting is the practice of selecting the column element with largest absolute value in the pivot column, and then interchanging the rows of the matrix so that this element is in the pivot position (the leftmost nonzero element in the row).

For example, in the matrix below the algorithm starts by identifying the largest value in the first column (the value in the (2,1) position equal to 1.1), and then interchanges the complete first and second rows so that this value appears in the (1,1) position.

The use of partial pivoting in Gaussian elimination reduces (but does not eliminate) roundoff errors in the calculation.

Reduced Row Echelon Form

A matrix is in row echelon form when:

• All nonzero rows are above rows of all zeros.

• The leading coefficient of each row is strictly to the right of the one in the row above it.

An example of a matrix in row echelon form is

$$A=\left(\begin{array}{ccc}1& 2& 3\\ 0& 4& 1\\ 0& 0& 2\end{array}\right).$$

An additional requirement for reduced row echelon form is:

• Every leading coefficient must be 1, and must be the only nonzero in its column.

While the identity matrix is most commonly associated with reduced row echelon form, other forms are possible. Another example of a matrix in reduced row echelon form is

$$A=\left(\begin{array}{cccc}1& 0& 0& 1\\ 0& 1& 0& 3\\ 0& 0& 1& -3\\ 0& 0& 0& 0\end{array}\right).$$

Algorithms

rref implements Gauss-Jordan elimination with partial pivoting. A default tolerance of max(size(A))*eps*norm(A,inf) tests for negligible column elements that are zeroed-out to reduce roundoff error.