Reduce Row Echelon Calculator

Reducing a matrix to reduced row echelon form is one of the most fundamental operations in linear algebra. It is the systematic procedure behind solving systems of linear equations, computing matrix inverses, determining rank, and analyzing vector spaces.

The Reduce Row Echelon Calculator on this page performs this for any 3×3 matrix. Enter your matrix, press Calculate, and get the RREF, rank, pivot columns, nullity, and whether it reduces to the identity matrix.

What Is Reduced Row Echelon Form?

A matrix is in RREF when: every leading entry (pivot) is 1; each leading 1 is the only nonzero entry in its column; leading 1s form a staircase; zero rows sit at the bottom. The process is Gauss-Jordan elimination using three row operations: swapping rows, scaling a row, adding a multiple of one row to another.

RREF is unique: every matrix has exactly one. A 3×3 matrix whose RREF is the identity is guaranteed invertible.

Why RREF Matters

RREF is the end state of the standard method for solving linear systems — solutions read directly with no back-substitution. A row like [0 0 0 | 5] signals no solution; zero rows reveal free variables. Beyond equations, RREF exposes rank (dimension of column/row space), pivot columns (basis for column space), and nullity (degrees of freedom). Rank + nullity = columns (rank-nullity theorem).

How to Use the Calculator

Step 1: Type your 3×3 matrix, one row per line, numbers separated by spaces (e.g., “1 2 3” / “0 1 4” / “5 6 0”).

Step 2: Verify 3 rows, 3 numbers each.

Step 3: Click Calculate.

Step 4: Read the RREF, Rank, Pivot Columns, Nullity, and Identity check.

Worked Example 1: Singular Matrix

Matrix: [1 2 3] [2 4 6] [1 1 1]. RREF: [1 0 -1] [0 1 2] [0 0 0]. Rank: 2. Pivots: 1, 2. Nullity: 1. Identity: No — row 2 was twice row 1, so singular.

Worked Example 2: Invertible Matrix

Matrix: [2 1 1] [1 3 2] [1 0 0]. RREF: identity [1 0 0] [0 1 0] [0 0 1]. Rank: 3. Nullity: 0. Identity: Yes — invertible, columns linearly independent.

Understanding Gauss-Jordan Elimination

The algorithm processes one column at a time: find a nonzero entry at/below the current row, swap it into position, scale to make the pivot 1, then clear every other entry in that column (above and below). The upward elimination distinguishes Gauss-Jordan from Gaussian elimination.

Pivot Columns, Rank, and Nullity

Pivot columns mark where independent information lives; matching columns of the original matrix form a basis for the column space. Rank = pivot count. Nullity = columns − rank = free variables. For 3×3: rank 3 = full rank and invertible; rank 2 = one dependency; rank 1 = all rows multiples of one row.

Tips

1. Enter rows exactly. Missing entries are the most common mistake.
2. Look for proportional rows first. Multiples mean rank below 3.
3. RREF is unique. Disagreement means arithmetic error — recheck.
4. Use the identity check as invertibility test.
5. Apply rank-nullity as sanity check. Rank + nullity = 3.

Frequently Asked Questions

1. What is RREF? The unique fully-simplified staircase form: pivots are 1, sole nonzeros in their columns, zero rows at bottom.

2. RREF vs REF? REF may have nonzeros above pivots; RREF clears them too and forces pivots to 1.

3. Three row operations? Swap rows, scale a row, add a multiple of one row to another.

4. RREF is identity — meaning? Matrix is invertible; determinant nonzero; columns independent.

5. Can two matrices share RREF? Yes — row-equivalent matrices share the unique RREF.

CONCLUSION

The calculator turns Gauss-Jordan elimination into one click, delivering RREF, rank, pivots, nullity, and invertibility. Because RREF is unique, the output is the definitive check for your hand calculations.