How to use it
- Type the matrix row by row, separating rows with a semicolon:
1 2 -1 -4; 2 3 -1 -11; -2 0 -3 22. - Pick fractions or decimals for the final matrix.
- Read the RREF, the rank and the pivot columns, and scroll through the row operations if you need the working.
How it works
The calculator runs Gauss–Jordan elimination column by column. In each column it finds the first row at or below the current one with a non-zero entry, swaps it up if needed (R1 ↔ R2), divides it so the pivot becomes 1 (R2 ← R2 ÷ 3) and subtracts multiples of it from every other row (R3 ← R3 − 2·R1) so the rest of the column becomes 0. Then it moves to the next row and column. All arithmetic is with exact fractions.
Worked example
The system x + 2y − z = −4, 2x + 3y − z = −11, −2x − 3z = 22 as an augmented matrix is 1 2 -1 -4; 2 3 -1 -11; -2 0 -3 22. The first steps are R2 ← R2 − 2·R1, giving [0 −1 1 −3], and R3 ← R3 + 2·R1, giving [0 4 −5 14]. After eight row operations the result is
[ 1 0 0 -8 ]
[ 0 1 0 1 ]
[ 0 0 1 -2 ]The rank is 3, the pivots are in columns 1, 2 and 3, and the solution reads straight off the last column: x = −8, y = 1, z = −2.
Tips and common mistakes
- Every row needs the same number of entries; write 0 for a missing term.
- Put the constants in the last column only for an augmented matrix; the calculator reduces every column the same way, so a pivot in the last column signals an inconsistent system.
- Row echelon form (REF) is not unique, but RREF is, so your answer should match exactly even if you used different steps.