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Linear Algebra

Reduced Row Echelon Form (RREF)

Reduced row echelon form (RREF) is the unique simplest form a matrix can be brought to by row operations. The calculator performs Gauss-Jordan elimination, showing every swap, scaling and elimination step.

Reduced Row Echelon Form (RREF)

Gauss-Jordan elimination with every row operation shown.

Try:
Answer[[1, 0, 0, -8], [0, 1, 0, 1], [0, 0, 1, -2]]
  1. Matrix A3×4 [[1, 2, -1, -4], [2, 3, -1, -11], [-2, 0, -3, 22]]
  2. MethodGauss-Jordan elimination — pivot, scale to 1, then clear above and below.
  3. Row opSwap R1 ↔ R2
  4. Row opR1 → (1/2)·R1
  5. Row opR2 → R2 − (1)·R1
  6. Row opR3 → R3 − (-2)·R1
  7. Row opSwap R2 ↔ R3
  8. Row opR2 → (1/3)·R2
  9. Row opR1 → R1 − (1.5)·R2
  10. Row opR3 → R3 − (0.5)·R2
  11. Row opR3 → (1/0.166667)·R3
  12. Row opR1 → R1 − (1.5)·R3
  13. Row opR2 → R2 − (-1.33333)·R3
  14. RREF[[1, 0, 0, -8], [0, 1, 0, 1], [0, 0, 1, -2]]
  15. Pivotscolumn 1, column 2, column 3
  16. Rank3
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Learn why this works

Vector Spaces, Span, and Basis RREF reveals span, rank, and basis of the column space →

Frequently asked questions

What makes RREF unique?

Each leading entry is 1, each leading 1 is the only non-zero entry in its column, leading 1s march down and to the right, and rows of zeros are at the bottom. These rules pin down the form uniquely.

Why is RREF useful?

It exposes the rank, identifies the pivot columns (a basis for the column space) and, for an augmented matrix, reads off the solution to Ax = b directly.

What's the difference between REF and RREF?

Row echelon form just requires zeros below each leading entry. RREF additionally requires the leading entry to be 1 and the only non-zero entry in its column.

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