RowEchelonForm

RowEchelonForm, commonly called row echelon form (REF), is a standard form of a matrix that can be obtained through a sequence of elementary row operations. A matrix is in row echelon form if all zero rows lie at the bottom, and the first nonzero entry of each nonzero row (the leading coefficient) is strictly to the right of the leading coefficient of the row above it. Additionally, every entry below a leading coefficient is zero.

In REF, the leading coefficients are not required to be equal to 1. If a matrix has the leading coefficient equal to 1 in each pivot position and zeros above and below each pivot, then it is in reduced row echelon form (RREF). RREF is a more refined and unique form, whereas REF is not unique.

The number of pivots in a matrix in REF equals its rank, the dimension of its row space (or column space). Any matrix can be transformed into REF by Gaussian elimination, but different sequences of row operations can produce different REF forms. REF is therefore a convenient intermediate form for understanding the structure of the matrix.

REF is particularly useful for solving linear systems. When applied to an augmented matrix, the presence and position of pivots reveal the consistency of the system and guide back-substitution to determine solutions. It helps identify whether a system has no solution, a unique solution, or infinitely many solutions, depending on the rank relative to the number of variables and augmented components.

Example: The matrix

[ [1, 2, 3],

[0, 4, 5],

[0, 0, 6] ]

is already in row echelon form, since the first nonzero entry in each row moves to the right and all entries below pivots are zero.