Definition
Elementary row operation
Swapping rows, rescaling one row by a nonzero scalar, or adding a multiple of another row.
An elementary row operation on a matrix over a field is one of three reversible operations: interchange two rows; multiply one row by a nonzero scalar; or add a scalar multiple of one row to a different row. Each is multiplication on the left by an invertible elementary matrix.
Linear systems
Applying the same row operation to both sides of preserves its solution set because it replaces the equation by with invertible. Applying an operation to without also changing generally changes the system. Row operations preserve rank but can change eigenvalues, since left multiplication need not be a change of basis by conjugation.