RREF Calculator

Reduced row echelon form is the unique simplest form of a matrix produced by row operations. In RREF, every pivot is 1, every pivot column has zeros elsewhere, and the pivots step down and to the right.

Rank counts the number of pivot rows in RREF. For a coefficient matrix, rank equals the number of independent equations. For an augmented system, comparing the rank of the coefficient matrix to the rank of the augmented matrix tells you whether the system is consistent.

Use 2x3 (for two equations in x and y) or 3x4 (for three equations in x, y, z). Enter each equation as one row: the coefficients of the variables in the left columns, and the right-hand side in the last column. The calculator reads the solution from RREF.

No solution: a row in RREF reads 0 = nonzero, which is impossible. Infinite solutions: the system has at least one free variable (a column without a pivot), so any value for that variable extends to a valid solution.

Reduced row echelon form is uniquely determined by the original matrix. Two different RREFs would imply contradictory pivot positions. This is what makes RREF useful for answering questions about a matrix at a glance.

Row echelon form (REF) only requires zeros below the pivots and a staircase shape. RREF goes further: pivots are exactly 1, and pivot columns have zeros above the pivot as well. Every matrix has a unique RREF.