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# Reduced Row Echelon Form of a Matrix (RREF) Calculator

This online calculator reduces given matrix to a reduced row echelon form (rref) or row canonical form and shows the process step by step.

This online calculator can help with RREF matrix problems. Not only it reduces given matrix into Reduced Row Echelon Form, but also shows the solution in terms of elementary row operations applied to the matrix. Definitions and theory can be found below the calculator

#### Reduced Row Echelon Form of a Matrix (RREF) Calculator

Reduced Row Echelon Form of a Matrix (RREF)

### Reduced Row Echelon Form of a Matrix

The matrix is said to be in Row Echelon Form (REF) if

• all nonzero rows (rows with at least one nonzero element) are above any rows of all zeroes
• 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 (although some texts said that the leading coefficient must be 1).

Example of matrix in REF form:

$\left[ \begin{array}{ccccc} 1 & 2 & 3 & 4 & 5 \\ 0 & 0 & 2 & 3 & 4 \\ 0 & 0 & 0 & 1 & 3 \end{array} \right]$

The matrix is said to be in Reduced Row Echelon Form (RREF) if

• it is in row echelon form
• the leading entry in each nonzero row is a 1 (called a leading 1)
• each column containing a leading 1 has zeros everywhere else

Example of matrix in RREF form:

$\left[{\begin{array}{ccccc}1&0&5&0&6\\0&1&5&0&6\\0&0&0&1&6\end{array}}\right]$

### Transformation to the Reduced Row Echelon Form

You can use a sequence of elementary row operations to transform any matrix to row echelon form and reduced row echelon form. Note that every matrix has a unique reduced row echelon form.

Elementary row operations are:

• Swapping two rows

$R_i \leftrightarrow R_j$.

• Multiplying a row by a non-zero constant

$kR_i \rightarrow R_i$

• Adding a multiple of one row to another row

$R_i+kR_j \rightarrow R_i$.

Elementary row operations preserve the row space of the matrix, so the resulting reduced row echelon matrix contains the generating set for the row space of the original matrix.

The calculator above shows all elementary row operations step-by-step, as well as their results, which are needed to transform given matrix to RREF.

PLANETCALC, Reduced Row Echelon Form of a Matrix (RREF) Calculator