Row Reduction (RREF)

Reduce a matrix to row echelon form, step by step

Matrix
rref(A) · Rank = 3 · Unique solution
Reduced row echelon form
10020103001−1
Rank
3
Nullity
0
Pivot columns
1, 2, 3
Free columns
none
Linear system
Unique solution
Steps
  1. 1.Scale rowR1 → 1/2·R1
    11/2−1/24−3−12−11−212−3
  2. 2.EliminateR2 → R2 + 3·R1
    11/2−1/2401/21/21−212−3
  3. 3.EliminateR3 → R3 + 2·R1
    11/2−1/2401/21/210215
  4. 4.Scale rowR2 → 2·R2
    11/2−1/2401120215
  5. 5.EliminateR1 → R1 − 1/2·R2
    10−1301120215
  6. 6.EliminateR3 → R3 − 2·R2
    10−13011200−11
  7. 7.Scale rowR3 → -1·R3
    10−130112001−1
  8. 8.EliminateR1 → R1 + 1·R3
    10020112001−1
  9. 9.EliminateR2 → R2 − 1·R3
    10020103001−1
Original matrix reduced to RREF with pivot positions highlightedThe input matrix on the left is transformed into its reduced row echelon form on the right; the leading 1 of each pivot column is highlighted.21−18−3−12−11−212−3rref10020103001−1