Reduced Row Echelon Form Calculator

What this calculator does

Solving a system of equations by hand is a careful business of adding and subtracting rows until the answer falls out. Row reduction is the organised way to do that, and the reduced row echelon form is its tidy end state, the point where the solution simply reads off. This takes a system of 2 or 3 equations and reduces it for you.

Enter a system that has a single solution and it returns the reduced form and that solution. If the equations do not pin down one answer, it tells you so. It runs right here in the browser.

Using the calculator

1. Choose whether to use the reduced form, which controls how far the reduction is taken.
2. Choose 2 or 3 equations.
3. Enter the coefficients and the constant for each equation, then press Calculate.

It works on the augmented matrix of your system, the grid of coefficients with the constants tacked on the right. Reset clears everything.

What reduced row echelon form is

Any matrix can be tidied up by juggling its rows, and reduced row echelon form, or RREF, is the tidiest it can get. A matrix is in RREF when three things hold: the first nonzero number in each row is a 1, called a leading 1 or pivot; each pivot is the only nonzero number in its whole column; and each pivot sits further to the right than the one in the row above, with any all-zero rows sitting at the bottom.

The payoff is that for a system of equations with a single answer, its RREF is about as simple as a grid can be, and the solution is sitting right there in it, as the next sections show.

The three moves that do not change the answer

Row reduction is built from just three moves, and the crucial thing is that none of them changes the solution of the system. You can always:

• Swap two rows.
• Multiply a whole row by a number (other than zero).
• Add a multiple of one row to another row.

Each of these is just a legal rewriting of the same set of equations, so the answers stay put. You keep applying them, aiming to create those leading 1s and clear out the columns around them, until the matrix is fully reduced.

Echelon, and fully reduced

There are two stopping points, and the toggle on this tool lets you pick between them. Row echelon form is the halfway house: a staircase of leading entries with zeros below each one, but not necessarily above. From there you can already work out the answer by back-substitution. Reduced row echelon form goes the rest of the way, cleaning out the entries above each pivot too and scaling every pivot to 1. The reduced form takes a little more work but leaves nothing left to do, which is why it is the more useful of the two.

How the reduced form hands you the solution

Here is the neat part. Take a system with a unique solution and reduce its augmented matrix fully. The coefficient part turns into the identity matrix, 1s down the diagonal and 0s everywhere else, and whatever is left in the final column is the answer. A row reading 1, 0, 3 simply says x equals 3. So once the matrix is in RREF, you are not solving anything more, you are just reading the values off. It is the same underlying job as Cramer's rule, reached by a different route, and the two always agree.

A worked example

Take the system x + y = 5 and x − y = 1. Its augmented matrix is [[1, 1, 5], [1, -1, 1]].

1. Subtract row 1 from row 2 to clear the first column below the pivot: [[1, 1, 5], [0, -2, -4]].
2. Divide row 2 by -2 to make its pivot a 1: [[1, 1, 5], [0, 1, 2]].
3. Subtract row 2 from row 1 to clear above that pivot: [[1, 0, 3], [0, 1, 2]].

The left side is now the identity, and the last column reads 3 and 2. So x = 3, y = 2, which you can check: 3 + 2 = 5 and 3 − 1... 3 − 2 = 1. Both hold.

Questions people ask

What is reduced row echelon form?

The simplest form of a matrix after row operations: each row's first nonzero entry is a 1, that 1 is alone in its column, and the pivots step to the right down the rows.

What is the difference between echelon and reduced echelon form?

Row echelon form has zeros only below each pivot, leaving back-substitution to do. Reduced form also has zeros above each pivot and scales every pivot to 1, so nothing is left to solve.

Which row operations are allowed?

Swapping two rows, multiplying a row by a nonzero number, and adding a multiple of one row to another. None of these changes the solution.

How does it solve a system?

Reduce the augmented matrix fully. The coefficient part becomes the identity, and the final column is the solution, ready to read off.

What if the system has no unique solution?

Then the reduced form will not turn into a clean identity. It will show a row of zeros for a dependent equation, or an impossible row for an inconsistent one, signalling infinitely many solutions or none.

References

A note on where this comes from. Bringing a matrix to reduced row echelon form is done by Gauss-Jordan elimination, a variation of Gaussian elimination named after Carl Friedrich Gauss and the geodesist Wilhelm Jordan, who described the reduced-form variant in the 1880s. (This is Wilhelm Jordan, not the mathematician Camille Jordan, a common mix-up.) For further reading, see Row echelon form.

1. Gauss-Jordan elimination, the procedure of applying row operations to bring a matrix to reduced row echelon form.
2. Wilhelm Jordan, the geodesist who published the reduced-form variant of Gaussian elimination, not to be confused with Camille Jordan.