A system of linear equations can be written as equations, as one vector equation, or as one matrix equation. For solving by hand the most useful form keeps only the numbers: the augmented matrix, one row per equation and one column per unknown.

A ledger writes numbers in fixed columns and never the words, because the column already says what each number means. An augmented matrix is the ledger of a system.

A system of $m$ equations in $n$ unknowns is $A\mathbf{x}=\mathbf{b}$, with the coefficients in the $m\times n$ matrix $A$ and the right-hand sides in $\mathbf{b}$. Its augmented matrix $[A\mid\mathbf{b}]$ is $A$ with $\mathbf{b}$ attached as an extra column behind a bar. Each row is one equation, each column before the bar belongs to one unknown, and the row operations of elimination act on the rows of this matrix.

• An unknown missing from an equation has the coefficient $0$ in its column; it is never simply left out.
• The operations $R_i\leftrightarrow R_j$, $R_i\to cR_i$ with $c\neq0$, and $R_i\to R_i+cR_j$ are the three moves of the lesson on elimination, written on the rows $R_i$ of the matrix instead of the equations $E_i$. They turn the augmented matrix of a system into that of an equivalent system, with exactly the same solutions.
• A system with $\mathbf{b}=\mathbf{0}$ is homogeneous. It always has the solution $\mathbf{x}=\mathbf{0}$, so the only question is whether it has others.
• The three readings say one thing: the equations, $x_1\mathbf{a}_1+\cdots+x_n\mathbf{a}_n=\mathbf{b}$, and .

Write the coefficients of each equation in a row, keeping every unknown in its own column, and put the right-hand side after the bar.

• A system is $A\mathbf{x}=\mathbf{b}$, and its augmented matrix is $[A\mid\mathbf{b}]$.
• Each row is an equation, each column before the bar an unknown; a missing unknown is a $0$.
• Row operations on the augmented matrix keep the solutions.
• A homogeneous system, $\mathbf{b}=\mathbf{0}$, always has the solution $\mathbf{0}$.

Great work! You now know how to:

• write a system as $A\mathbf{x}=\mathbf{b}$ and as an augmented matrix;
• read a system back from its augmented matrix;
• apply row operations to an augmented matrix;
• recognise a homogeneous system and its obvious solution.