Row Echelon Form (REF) Calculator
This row echelon form calculator reduces a matrix to row echelon form with steps — every elementary row operation named and the matrix redrawn after each one — and then stops. If your exercise asks for the echelon form, that is what you get, not the reduced one.
Take 1 2 3 / 4 5 6 / 7 8 9. Subtract 4 times the first row from the second, 7 times it from the third, tidy the second row, and clear below the new pivot. What comes out is 1 2 3 / 0 1 2 / 0 0 0 — two pivots and one zero row, which is the whole reason that particular matrix turns up in every textbook.
What Row Echelon Form Means
Three conditions, and that is all:
1. Any row of all zeros sits at the bottom. 2. The first non-zero entry of each row — its pivot, or leading entry — lies strictly to the right of the pivot in the row above. 3. Everything below a pivot is zero.
Together those give the staircase of zeros the form is named for. On a square matrix it comes out looking upper triangular, but the idea is wider than that: a staircase works on any shape, so a 2×3 or a 4×3 has a row echelon form just as a 3×3 does.
The Three Row Operations, and How to Use Them
Only three moves are allowed, and they are called the elementary row operations: swap two rows, multiply a row by a non-zero number, and add a multiple of one row to another. Nothing else. Each of them leaves the row space alone, which is why the matrix you finish with still says exactly what the one you started with said.
The procedure is called forward elimination, and it goes column by column from the left. Find a non-zero entry at or below the current row and make it the pivot, swapping rows if you have to. Then use it to clear everything underneath. Move one row down and one column right, and repeat. If a whole column is zeros, step past it and carry on.
For 0 2 3 / 1 4 5 / 2 8 10 the first entry is a zero, so the first move has to be a row swap: R1 ⇄ R2. Then R3 − 2R1 clears the bottom of the first column, and the answer is 1 4 5 / 0 1 3/2 / 0 0 0. Three operations, and the calculator shows the matrix as it stands after each one.
This is the same idea you met when solving equations by cancelling one letter at a time — the elimination method calculator does it in the language of equations, while this page does it in the language of a grid. Row reduction is also the second way to reach a determinant: reduce a square matrix, and if a zero row appears, its determinant is zero and it has no inverse.
Why Your Answer Can Look Different and Still Be Right
This is the part that costs students time, and most calculators say nothing about it. Row echelon form is not unique. A different order of operations gives a different matrix, and both are correct echelon forms of the same thing.
Our own two settings prove it. That first example, with leading entries forced to 1, gives 1 2 3 / 0 1 2 / 0 0 0. With leading entries left alone it gives 1 2 3 / 0 −3 −6 / 0 0 0. Different middle row, same matrix, and neither is wrong.
But two things can never differ, whatever route you take:
The pivots land in the same columns, and the number of zero rows is the same. In both versions above the pivots sit in columns 1 and 2, and there is exactly one zero row. So when your working looks nothing like the answer in the back of the book, check those two first. If they match, you have not made a mistake — you took a different road to a different-looking but equally valid answer. If they do not match, one of you has gone wrong, and now you know where to look.
Carrying on past this point — clearing above every pivot as well — leads to a form with a different property: that one is unique, so there a difference really would mean a mistake. It is a separate exercise with its own page, and if that is what your question asked for, the reduced row echelon form calculator does the second pass as well. If your question asked for the echelon form, this page has already given you the answer.
Do Leading Entries Have to Be 1?
Textbooks disagree, and both camps are large. Some definitions insist every leading entry is a 1; others accept any non-zero number and only care about the staircase. Neither is more correct, but handing in the wrong one can still cost you marks, so this calculator has both and you can switch between them and watch the answer change.
Turning leading ones on adds a division for every pivot row, which is why the second version of the example above has −3 and −6 where the first has 1 and 2. If your course does not ask for them, leaving them off usually keeps the numbers whole and the arithmetic shorter.
How to Use This 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 fine — a 2×3 reduces just as happily as a 3×3.
2. Choose the convention. The two buttons above the grid pick whether leading entries must be 1 or can be any number. Change it after an answer is showing and the whole thing is worked again.
3. 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, blanks count as zero, and every operation is worked in exact fractions.
4. Press = and open the working. Show every row operation gives one block per move, each headed the way you would write it in an exercise book — R1 ⇄ R2, R2 ÷ 3, R3 − 2R1 — with the matrix as it stands after that move. The last block marks the pivots.
Every key and control, in one place:
| rows / columns | Sets the size, 1 to 4, each one on its own; rectangles are welcome |
| leading entries | Switches between the two textbook conventions and re-works the answer |
| 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 |
| My keyboard | Switches that cell to your own keyboard, for anything the pad does not carry |
| copy | The icon in the band copies the finished form, row by row |
One quiet detail worth knowing: where there is a choice of pivot, this calculator takes the friendliest number rather than the largest one. Numerical software picks the largest to protect against rounding, but nothing here is rounded, so a 1 is chosen when there is one going spare and the fractions stay small enough to follow by hand.
Row Echelon Form FAQ
What is row echelon form?
Does row echelon form mean clearing above the pivots too?
Why does my row echelon form look different from the calculator's?
Do the leading entries have to be 1 in row echelon form?
Can a rectangular matrix be put in row echelon form?
What are the three elementary row operations?
Clear below each pivot, keep the staircase stepping right, and stop when it does. If your answer differs from the book, count the pivots and the zero rows before you assume the worst — more often than not, both of you are right.