Row Operations in Action

Linear Algebra (Matrices) · Axiom Academy

EXAMPLE Row Operations in Action Reduce an augmented matrix to row echelon form one elementary row operation at a time. Use elementary row operations to transform the augmented matrix below to row echelon form (each leading entry a 1 , with zeros beneath every pivot). It represents the linear system 2x + y + 3z = 8 , , . Every row operation keeps the system — and its solution — unchanged. Nice work! You reduced the matrix to row echelon form one operation at a time. Here is what that practiced: Leading 1s first: scale a row (like ) to turn each pivot into a 1 . Eliminate downward: use row replacement ( ) to make zeros below each pivot. Column by column: finish all of column 1 before starting column 2, then column 3. Solutions are preserved: every elementary operation is reversible, so the system stays equivalent — here (x, y, z) = (0, -1, 3) throughout. This systematic row reduction is the engine behind solving linear systems, inverting matrices, and computing rank. Practice on new matrices to make the strategy automatic.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.