Elementary Row Operations in Matrix Theory constitute a fundamental mechanism within linear algebra defined by three specific transformations: row swapping (permutation), scalar multiplication of a single row, and the addition of a multiple of one row to another. These operations form an equivalence relation that preserves matrix rank while altering determinant values according to precise rules regarding sign changes and scaling factors. As elementary generators for Gaussian elimination and LU decomposition, they serve as the canonical method for reducing matrices to echelon forms without changing the linear dependency structure of the underlying vector space.
Prerequisite chain context: requires Homogeneous Systems of Linear Equations in Math.