How this calculator works
Gaussian elimination chooses a pivot in each column, scales it to one, and clears the entries below it to reach row echelon form. Gauss-Jordan elimination also clears the entries above each pivot to reach reduced row echelon form, abbreviated RREF. Both forms preserve the solution set of an augmented system, while RREF makes the leading variables immediately readable. The calculator uses partial pivoting, so it may reorder rows to keep the largest available pivot. This page covers both operations, including the extended matrix technique often called Gauss-Jordan elimination.
Formula
Row operations are swapping two rows, multiplying a row by a non-zero number, and adding a multiple of one row to another. RREF has leading ones with zeros above and below each leading one.
Worked example
The matrix with rows 1, 2, 3 and 0, 0, 1 is already in row echelon form with pivots in columns 1 and 3. Its reduced form clears the entry 3 above the second pivot and gives 1, 2, 0; 0, 0, 1.
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Frequently asked questions
Is this the same as Gauss-Jordan elimination?
Reduced row echelon form is the result of Gauss-Jordan elimination. The page also offers plain row echelon form, which stops after forward elimination.
Can I solve a system with it?
Yes. Append the right-hand side as an extra column and reduce the augmented matrix. A pivot in the extra column means no solution; fewer pivots than unknowns means infinitely many.
Why do zero rows appear?
Zero rows record dependent equations. They do not change the solution set, which is why the rank counts only the pivot rows.