elementary row operation

An elementary row operation is one of three small, legal moves you can make on the rows of a matrix without changing the solution it represents. Think of them as the only allowed moves in a puzzle: each one rearranges things while preserving what really matters.

There are exactly three. First, swap two rows (interchange their positions). Second, scale a row by any nonzero number (multiply every entry in it by that number). Third, add a multiple of one row to another row. Notice the “nonzero” in scaling — multiplying a row by zero would destroy information and is forbidden.

Each operation mirrors a perfectly valid manipulation of the underlying equations: reordering equations, multiplying an equation through by a constant, or adding equations together. Applying these operations in sequence — Gaussian elimination — drives a matrix toward echelon form and ultimately reads out the system's solution.