Row Operations

Principle

An augmented matrix stores the coefficient matrix and the right-hand side of a linear system together. For a system \(A\mathbf{x}=\mathbf{b}\), the augmented matrix is written \((A\mid\mathbf{b})\).

Row operations change the written form of the system without changing the set of solutions. They are the matrix version of rearranging, rescaling, and combining equations.

Notation

• \(R_i\leftrightarrow R_j\) means swap row \(i\) and row \(j\).
• \(cR_i\) means replace row \(i\) by \(c\) times row \(i\), where \(c\ne0\).
• \(R_i+cR_j\) means replace row \(i\) by row \(i\) plus \(c\) times row \(j\).
• \((A\mid\mathbf{b})\) means the coefficients \(A\) and constants \(\mathbf{b}\) are stored in one augmented matrix.

Method

To use row operations on an augmented matrix:

1. Write every equation in the same variable order.
2. Put coefficients on the left of the vertical bar and constants on the right.
3. Choose a row operation that simplifies the matrix.
4. Apply the operation to the whole row, including the augmented column.
5. Translate the final rows back into equations when needed.

Rules

• Swapping two rows preserves the solution set because it only changes the order of the equations.
• Multiplying a row by \(c\ne0\) preserves the solution set because an equation and a nonzero multiple of that equation have the same solutions.
• Replacing \(R_i\) by \(R_i+cR_j\) preserves the solution set because any solution satisfying both original equations also satisfies their linear combination, and the step can be undone by subtracting \(cR_j\).
• Scaling by zero is not allowed because it can erase an equation and add false solutions.
• Row operations must act on complete rows, not only selected entries.

Examples

The system

\[ \begin{aligned} x+y&=5,\\ 2x+3y&=13 \end{aligned} \]

has augmented matrix

\[ \left(\begin{array}{cc|c} 1&1&5\\ 2&3&13 \end{array}\right). \]

Use \(R_2-2R_1\), which means replace row \(2\) by row \(2\) plus \(-2\) times row \(1\):

\[ \left(\begin{array}{cc|c} 1&1&5\\ 2&3&13 \end{array}\right) \longrightarrow \left(\begin{array}{cc|c} 1&1&5\\ 0&1&3 \end{array}\right). \]

The new second row represents \(y=3\), and the first row remains \(x+y=5\). The operation simplified the system without changing its solutions.

Checks

• Check that the augmented matrix includes both the coefficient columns and the right-hand-side column.
• Check that each row operation is applied to the entire augmented row, including entries after the vertical bar.
• Check that a scaling operation never uses \(c=0\).
• Check that the operation you write matches the row you actually replace.