Use this calculator to transform a matrix into row canonical form. This is also called reduced row echelon form (RREF). The theory is explained at Transforming a matrix to reduced row echelon form.

A matrix of any size may be entered using integer or rational numbers. Valid number formats are "3", "-3", "3/4" and "-3/4". (Real numbers are not supported, so decimal points "." will be ignored.)

Copy and paste one of the following matrices (the yellow ones on the left) into the box above to test. The solution is shown on the right.

Example matrix | → | Row canonical form |
---|---|---|

1, -2, 3, 1, 2 1, 1, 4, -1, 3 2, 5, 9, -2, 8 |
→ |
1 0 11/3 0 17/6 0 1 1/3 0 2/3 0 0 0 1 1/2 |

0 2 2 1/3 0 -5 10 5/3 -3 6 0 -1 |
→ |
1 0 0 1/3 0 1 0 0 0 0 1 1/6 |

0, 2, 2, -1, 6, 4 0, 4, 4, 1, 10, 13 0, 8, 8, -1, 26, 23 |
→ |
0 1 1 0 0 3/2 0 0 0 1 0 2 0 0 0 0 1 1/2 |

3, 6, 3, -7 1, 2, -1, 3 2, 4, 1, -2 |
→ |
1 2 0 1/3 0 0 1 -8/3 0 0 0 0 |

1,2,-3,1,2 2,4,-4,6,10 3,6,-6,9,13 |
→ |
1 2 0 7 0 0 0 1 2 0 0 0 0 0 1 |

835/1000 667/1000 168/1000 333/1000 266/1000 67/1000 |
→ |
1 0 1 0 1 -1 |

1 5 2 17 0 2 2 10 0 0 5 15 |
→ |
1 0 0 1 0 1 0 2 0 0 1 3 |

1 2 1 3 2 5 -1 -4 3 -2 -1 5 |
→ |
1 0 0 2 0 1 0 -1 0 0 1 3 |

1 2 1 1 0 0 2 5 -1 0 1 0 3 -2 -1 0 0 1 |
→ |
1 0 0 1/4 0 1/4 0 1 0 1/28 1/7 -3/28 0 0 1 19/28 -2/7 -1/28 |

Note that the last example shows how to invert the square matrix `A`. The last but one example shows how to
solve the equation `Ax = b`

.

What is the RREF of the square matrix `A`?
Is this the case for all square invertible matrices? Go on, try it.

A matrix `A` is said to be in **row canonical form** (or **reduced row echelon form** (RREF))
if the following conditions hold
(where a leading nonzero element of a row of `A` is the first nonzero element in the row):

- All zero rows, if any, are at the bottom of the matrix.
- Each leading nonzero entry in a row is to the right of the leading nonzero entry in the preceding row.
- Each pivot (leading nonzero entry) is equal to 1.
- Each pivot is the only nonzero entry in its column.

**Theorem** Every matrix A is row equivalent to a unique matrix in row canonical form.

More on the theory at Transforming a matrix to reduced row echelon form.

**[LIPS08]**Lipschultz and Lipson, Schaum's Outline of Linear Algebra, Schaum's Outlines, 4th edition, 2008.

See also our coding theory matrix calculator which transforms a generator matrix or parity-check matrix of a linear [n,k]-code into standard form.

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*Last updated: 13 August 2020 10:13Z*