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In linear algebra a matrix is in row echelon form if

All nonzero rows (rows with at least one nonzero element) are above any rows of all zeroes, and
The leading coefficient (the first nonzero number from the left, also called the pivot) of a nonzero row is always strictly to the right of the leading coefficient of the row above it.

Some texts add a third condition:

The leading coefficient of each nonzero row is 1.^[1]

A matrix is in reduced row echelon form (also called row canonical form) if it satisfies the additional condition:

Every leading coefficient is 1 and is the only nonzero entry in its column.

The first nonzero entry in each row is called a pivot.

[edit] Examples

This matrix is in reduced row echelon form:

$\begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \end{bmatrix}$

The following matrix is also in row echelon form, but not in reduced row form:

$\begin{bmatrix} 1 & 9 & 1 & 1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 3 \\ \end{bmatrix}$

However, the matrix below is not in row echelon form, as the leading coefficient of row 3 is not strictly to the right of the leading coefficient of row 2, and the main diagonal is not made up of only ones:

$\begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 3 & 7 & 2 \\ 0 & 6 & 0 & 0 \\ \end{bmatrix}$

[edit] Non-uniqueness

Every nonzero matrix can be reduced to an infinite number of echelon forms (they can all be multiples of each other, for example) via elementary matrix transformations. However, all matrices and their row echelon forms correspond to exactly one matrix in reduced row echelon form.

[edit] Systems of linear equations

A system of linear equations is said to be in row echelon form if its augmented matrix is in row echelon form. Similarly, a system of equations is said to be in reduced row echelon form or canonical form if its augmented matrix is in reduced row echelon form.

[edit] Pseudocode

The following pseudocode converts a matrix to reduced row-echelon form:

 function ToReducedRowEchelonForm(Matrix M) is     lead := 0     rowCount := the number of rows in M     columnCount := the number of columns in M     for 0 ≤ r < rowCount do         if columnCount ≤ lead then             stop         end if         i = r         while M[i, lead] = 0 do             i = i + 1             if rowCount = i then                 i = r                 lead = lead + 1                 if columnCount = lead then                     stop                 end if             end if         end while         Swap rows i and r         Divide row r by M[r, lead]         for 0 ≤ i < rowCount do             if i ≠ r do                 Subtract M[i, lead] multiplied by row r from row i             end if         end for         lead = lead + 1     end for end function

[edit] See also

[edit] Notes

^ See, for instance, Larson and Hostetler, Precalculus, 7th edition.

[0] See, for instance, Larson and Hostetler, Precalculus, 7th edition.

[1]