Row Echelon Form Calculator

How it is calculated

Enter a 3×3 matrix row by row (a11 to a33). The tool does forward Gaussian elimination column by column. In each column it looks at the entries on and below the current row and swaps the one with the largest absolute value into the pivot position (partial pivoting). It then subtracts factor × pivot row from the rows below, where factor = entry ÷ pivot, to make those entries zero. A column with no usable pivot (all entries below 10⁻¹²) is skipped. The rank is the number of pivots found. The determinant is the product of the diagonal of the final matrix times −1 for each row swap. Pivots are not scaled to 1, so this is a row echelon form, not the reduced one. Entries are shown to four decimals.

Common mistakes

• Expecting leading ones. The tool leaves pivots as they are, so a leading −3 does not get divided to 1; dividing each row by its pivot, then clearing above, is a separate step towards reduced form.
• Expecting the original row order. Partial pivoting swaps rows, so the first row of the result may be a different row from the one you typed first.
• Reading the matrix as an augmented system and treating the last column like constants. The tool treats all nine entries as a plain 3×3 matrix and gives no solution values.
• Trusting rank from rounded decimals. A pivot smaller than 10⁻¹² is treated as zero, so typing 0.3333 for ⅓ can give a full rank where the exact fractions would give less.
• Comparing the determinant directly with a hand answer that skipped the sign. Each row swap flips the sign, and the tool includes this.