What is Row Echelon Form?
Row Echelon Form (REF) is a particular form of a matrix achieved through Gaussian elimination. A matrix is in row echelon form when it satisfies three conditions: all zero rows are at the bottom, the leading entry (pivot) of each nonzero row is to the right of the pivot of the row above it, and all entries in a column below a pivot are zeros.
Reduced Row Echelon Form (RREF)
RREF goes further by requiring that each pivot is 1 and is the only nonzero entry in its column. RREF is unique for any given matrix and directly reveals the solutions to a system of linear equations.
Elementary Row Operations
Row Swap
Exchange two rows of the matrix.
R_i ↔ R_j
Row Scaling
Multiply a row by a nonzero scalar.
R_i → k * R_i
Row Replacement
Add a scalar multiple of one row to another.
R_i → R_i + k * R_j
Applications
- Solving linear systems: Row reduction transforms augmented matrices to find solutions.
- Finding matrix rank: The number of pivots equals the rank.
- Computing determinants: Row operations simplify determinant calculation.
- Finding inverse matrices: Augmenting with identity and reducing to RREF yields the inverse.