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# rref

Reduced row echelon form

## Syntax

R = rref(A)
[R,jb] = rref(A)
[R,jb] = rref(A,tol)

## Description

R = rref(A) produces the reduced row echelon form of A using Gauss Jordan elimination with partial pivoting. A default tolerance of (max(size(A))*eps *norm(A,inf)) tests for negligible column elements.

[R,jb] = rref(A) also returns a vector jb such that:

• r = length(jb) is this algorithm's idea of the rank of A.

• x(jb) are the pivot variables in a linear system Ax = b.

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

• R(1:r,jb) is the r-by-r identity matrix.

[R,jb] = rref(A,tol) uses the given tolerance in the rank tests.

Roundoff errors may cause this algorithm to compute a different value for the rank than rank, orth and null. Additionally, use mldivide to solve linear systems when high precision is required.

## Examples

Use rref on a rank-deficient magic square:

```A = magic(4), R = rref(A)

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

R =
1    0    0    1
0    1    0    3
0    0    1   -3
0    0    0    0```