How the RREF calculator works
Row reduction, also called Gauss-Jordan elimination, turns any matrix into its reduced row echelon form using only the three elementary row operations. The calculator works column by column: it finds a nonzero entry to serve as the pivot, swaps it into position if needed, scales the row so the pivot becomes 1, and then subtracts multiples of that row from every other row to zero out the rest of the column. The row operations are listed in order with a snapshot of the matrix after each one, so you can follow or check your own work line by line.
Worked example: the default 3x4 augmented matrix
The default matrix encodes the system 2x + y - z = 8, -3x - y + 2z = -11, -2x + y + 2z = -3. The first pivot is the 2 in row 1, so row 1 is scaled by 1/2 to get a leading 1. Adding 3 times row 1 to row 2 and 2 times row 1 to row 3 clears the first column. The second column's pivot is now the 1/2 in row 2; scaling it to 1 and clearing the column above and below leaves a third pivot in row 3, which is cleared the same way. The final matrix has the identity on the left and the constants 2, 3, -1 on the right: x = 2, y = 3, z = -1. Because every variable column has a pivot, the solution is unique.
Reading the result
| What you see | Meaning |
|---|---|
| Rank | The number of pivots, which is the number of independent rows (and columns). |
| Pivot columns | Columns containing a leading 1. In an augmented system these are the determined variables. |
| Free variables | Variable columns without a pivot. Each one adds a parameter to the solution. |
| Row 0 0 0 | c with c not 0 | The system is inconsistent and has no solution. |
| Row of all zeros | That equation was a combination of the others and carries no new information. |
The three row operations
| Operation | Notation in the steps | Effect on a determinant |
|---|---|---|
| Swap two rows | Swap R1 and R2 | Changes sign |
| Scale a row by k | R2 -> (1/3) R2 | Multiplies by k |
| Add a multiple of a row | R3 -> R3 - 2R1 | No change |
Why RREF matters
The reduced form answers most first questions about a matrix at once. Its pivot count is the rank, its non-pivot columns describe the null space, and when the matrix is augmented with a right-hand side it displays the solution of the linear system directly. Row reducing [A | I] produces [I | A^-1] when A is invertible, which is how the matrix calculator on this site computes inverses. Unlike row echelon form, which depends on the order of operations, RREF is unique, so two people reducing the same matrix by different paths must arrive at the same result.
Tips for entering matrices
Cells accept integers, decimals, and fractions written as p/q. Decimals are converted to exact fractions, so 0.25 becomes 1/4. Use the size selectors for anything up to 6 rows by 7 columns. For a system of n equations in n unknowns use n rows and n + 1 columns with the augmented box ticked; untick it to reduce a plain matrix where every column may hold a pivot. You can share a filled-in matrix with a link of the form ?m=1,2;3,4, one row per semicolon.