Systems and Augmented Matrices

A linear system is a set of equations in which every unknown appears alone, never squared and never multiplied by another unknown.

The letters are decoration

Only the coefficients and the constants matter. Writing them in a grid loses nothing and saves a great deal of copying.

The augmented matrix

The coefficients form a matrix and the constants form one extra column. A vertical rule separates them, marking [A|b].

Why the rule matters

The last column is not another coefficient. It is what the equations must equal, and the rule keeps that distinction visible.

Three legal moves

Swap two rows, multiply a row by a non-zero number, or add a multiple of one row to another. Each is reversible.

Why they are safe

Every row operation is something you could do to the equations themselves. The solution set never changes.

Strip the letters, keep the numbers

An augmented matrix records the coefficients and the constants of a system, with a bar separating them. The letters carry no information beyond position, so removing them loses nothing.

Rows are equations, columns are variables

Each row is one equation and each column one unknown. Keeping that correspondence in mind is what makes row operations interpretable rather than mechanical.

The three row operations

Swap two rows, multiply a row by a nonzero scalar, add a multiple of one row to another. Each corresponds to a legitimate manipulation of equations, which is why the solution set is preserved.

Row operations preserve solutions

Every row operation is reversible, so the new system has exactly the same solutions as the old. That is the entire justification for elimination, and it is worth stating rather than assuming.