Reduced Row Echelon Form (RREF) Calculator
This reduced row echelon form calculator takes a matrix all the way to RREF with steps, and it shows the working in two labelled halves — the forward pass down, then the back pass up — because leaving the second one out is the commonest way this goes wrong.
Take 1 2 3 / 4 5 6 / 7 8 9. The forward pass gives 1 2 3 / 0 1 2 / 0 0 0. One more operation, R1 − 2R2, clears the entry above the second pivot, and the reduced form is 1 0 −1 / 0 1 2 / 0 0 0. Two pivots, one zero row, and — this is the part worth holding on to — that is the only answer there is.
What Makes a Matrix Reduced
Four conditions, all of which have to hold at once:
1. Any row of all zeros sits at the bottom. 2. Every leading entry — the first non-zero number in a row, called the pivot — is a 1. 3. Each pivot lies strictly to the right of the pivot in the row above, giving the staircase. 4. A pivot is the only non-zero entry in its whole column, above it as well as below.
The fourth is the one that does the real work. Without it you have an ordinary echelon form; with it you have the reduced one. This calculator ticks all four off against the answer before showing it to you, rather than simply asserting that the answer is right.
The Two Passes of Gauss-Jordan
The method has two halves, and this page keeps them apart on purpose.
The forward pass works down the matrix, column by column. Find a pivot, divide its row through so the pivot becomes 1, and clear everything below it. For our example that is R2 − 4R1, then R3 − 7R1, then R2 ÷ (−3), then R3 + 6R2. What comes out, 1 2 3 / 0 1 2 / 0 0 0, is the halfway point.
That halfway point has a name of its own: it is the row echelon form. Plenty of exercises ask for nothing more, and if yours is one of them you are already finished — the row echelon form calculator stops exactly there and shows the same forward pass on its own, including the version where leading entries are left as they are.
The back pass starts at the bottom pivot and works upwards, clearing the entries above each one. In our example a single operation does it, R1 − 2R2. In a bigger case there are more: reducing 2 1 −1 / −3 −1 2 / −2 1 2 takes seven operations down and three back up, and lands on the identity matrix.
Why There Is Only One Right Answer
Here is the property that makes this form worth the extra work. Every matrix has exactly one reduced row echelon form. Do the operations in a different order, pick different pivots, take a longer route — you will still land on the same matrix.
That has a blunt practical consequence. If your answer and this one differ, you have not simply taken another road: one of the two is wrong. It is worth knowing that, because the unreduced echelon form behaves the opposite way, and students who have been told “there are many correct answers” often carry that belief into this exercise where it does not apply.
The same uniqueness explains why this form is used to settle other questions. A square matrix reduces to the identity matrix if and only if it can be inverted, which is the same as saying its determinant is not zero; when it does not reduce to the identity, no inverse exists. One canonical form, several questions answered.
Common Mistakes in Gauss-Jordan Elimination
Four slips account for almost every wrong answer, and the layout of this page is built around them.
Stopping after the forward pass. The matrix looks finished — it has its staircase and its zeros below — but the entries above the pivots are still there. That is why the two passes are labelled separately here instead of running together.
Sign errors. Subtracting a negative multiple is where most of them happen, so every operation is written out in full, R3 + 6R2 rather than a vague “add row 2”.
Dropped fractions. Dividing a row by 3 turns clean numbers into thirds, and rounding one of them to 0.33 quietly ruins everything downstream. Nothing here is rounded at any point.
Treating a zero as a pivot. If the entry you want is zero, you must swap a row up instead. The calculator does the swap and names it, so you can see where it happened.
How to Use This Reduced Row Echelon Form Calculator
The screen has three parts: the answer band at the top, the matrix you fill in below it, and the keypad underneath.
1. Set the size. Rows and columns are chosen separately, from 1 to 4, and any shape is allowed — a 2×3 reduces just as a 3×3 does.
2. Fill in the cells. Tap a cell and use the keypad. Whole numbers, decimals such as 0.25 and fractions such as 3/4 are all accepted, and a blank counts as zero.
3. Press =. The reduced form appears at the top with its pivots picked out, and the line beneath counts the pivots and the zero rows.
4. Open both passes. Show both passes lays out the working: a heading for pass one, every operation with the matrix after it, the echelon milestone in between, a heading for pass two, and finally the four conditions ticked off against the answer.
Every key and control, in one place:
| rows / columns | Sets the size, 1 to 4, each one on its own; rectangles are welcome |
| the cells | Tap to select; blank counts as zero; fractions and decimals both accepted |
| ← → | Move to the previous or next cell, wrapping round the whole grid |
| ↑ ↓ | Move up or down a row, for filling a column at a time |
| ± | Flips the sign of the cell you are in, so a minus cannot be typed twice by accident |
| ⁄ | The fraction bar. 3⁄4 stays a fraction all the way through |
| . | Decimal point. 0.25 is read as one quarter, exactly |
| ⌫ | Deletes one character from the cell you are in |
| AC | Clears the whole matrix and starts again |
| = | Reduces it all the way |
| My keyboard | Switches that cell to your own keyboard, for anything the pad does not carry |
| copy | The icon in the band copies the reduced form, row by row |
Change anything after an answer is showing and the band dims and asks you to press = again, so what you are reading always belongs to what is in the cells. There are no limits, no ads and no account.
Reduced Row Echelon Form FAQ
What is reduced row echelon form?
Is the reduced row echelon form of a matrix unique?
What are the two passes of Gauss-Jordan elimination?
Why is my RREF different from the calculator's?
Can a rectangular matrix have a reduced row echelon form?
Does this RREF calculator show the steps?
Down to the staircase, then back up to clear above every pivot, and check the four conditions at the end. Do that and you have the one form the matrix was always going to give you — which is exactly why it is worth doing properly.