RREF Calculator

What reduced row echelon form means

A matrix is in reduced row echelon form when each nonzero row has a leading 1, each leading 1 is farther right than the leading 1 above it, every other entry in a pivot column is zero, and any all-zero rows are at the bottom. The leading 1s are the pivots.

RREF gives a consistent way to expose the numerical structure of a matrix. The pivot columns identify where pivots occur, while the remaining columns are free columns. The rank is the number of pivots. For a matrix with n columns, the nullity reported here is n minus the rank, which is also the number of free columns.

This tool numbers columns starting at 1. If it reports pivot columns 1 and 3, those are the first and third columns of the matrix you entered.

The row operations shown

The calculator uses the three standard kinds of elementary row operation:

• Swap two rows.
• Multiply a row by a nonzero number.
• Replace one row with itself minus a multiple of another row.

For each pivot column, it chooses the remaining row with the largest absolute pivot candidate. This partial-pivoting step is a numerical safeguard: it avoids dividing by a smaller available value. The chosen row is moved into place, its pivot is scaled to 1, and entries above and below that pivot are eliminated. Every resulting snapshot appears in the elementary row operations list.

Different valid sequences of row operations may look different along the way. The final reduced row echelon form is the same apart from small effects of finite-precision arithmetic.

Worked numeric example

For the default matrix

the reduced row echelon form is

There are two pivots, in columns 1 and 3, so the rank is 2. Column 2 is free, and the nullity is 3 - 2 = 1. The calculator displays the exact row-operation sequence it used to reach this result rather than presenting only the final matrix.

Rank, nullity, pivot columns, and free columns

Rank counts the pivot positions in the reduced matrix. It describes the dimension of the matrix’s column space and also equals the number of nonzero rows in RREF.

Nullity is the dimension of the null space of the matrix. For an m × n matrix with rank r, the rank-nullity relationship is:

rank + nullity = number of columns, or r + nullity = n.

The calculator determines these values directly from the numeric reduction. A matrix may have no free columns, in which case its nullity is zero. A wide rectangular matrix can have several free columns, and a matrix with dependent or zero rows can have rank below its row count.

Input rules and numerical tolerance

Enter one row per line. Spaces, commas, semicolons, and vertical bars can separate entries on the English page. Blank lines are ignored, but nonblank rows must all have equal length. Values can be integers, decimals, or scientific notation such as 2.5e-3. Each value must be finite and between -1,000,000 and 1,000,000.

The calculator uses floating-point arithmetic. During reduction, values with absolute size below 1e-10 are treated as zero. That tolerance prevents tiny rounding remnants from appearing as false pivots, but it also means results near numerical singularity should be checked with the precision requirements of your course or application.

What this calculator does not do

This is a numeric matrix reducer, not a symbolic algebra system. It does not accept variables, fractions written as expressions, radicals, or formulas. It does not calculate a determinant or inverse, and it does not produce a parametric solution, translate rows into equations, or decide whether a separately described system is consistent.

You may enter a numeric augmented matrix to inspect its row reduction, but the tool treats every column as part of one matrix. It does not recognize an augmentation boundary or interpret the last column as constants. Use the displayed RREF and column information as matrix output, then apply the conventions required by your own linear algebra problem.