Definition For a system of equations in row echelon form (which may be represented by an augmented matrix), a variable whose coefficient is a leading 1 (pivot) is called a basic variable, and a variable without a pivot is called a free variable.
Consider the following system of equations, already in row echelon form:
\[ \begin{cases} x_1 + 2x_2 - x_3 + x_4 - 3x_5 = 0 \\ \phantom{x_1 + 2x_2} x_3 - 2x_5 = 0 \\ \phantom{x_1 + 2x_2 + x_3} x_4 - x_5 = 0 \end{cases} \]The corresponding augmented matrix in row echelon form is:
\[ \left[ \begin{array}{ccccc|c} 1 & 2 & -1 & 1 & -3 & 0 \\ 0 & 0 & 1 & 0 & -2 & 0 \\ 0 & 0 & 0 & 1 & -1 & 0 \end{array} \right] \]The pivot columns are columns 1, 3, and 4. Hence:
Solving for the basic variables in terms of the free ones:
Substituting \( x_3 = 2x_5 \) and \( x_4 = x_5 \):
\[ x_1 = -2x_2 + 2x_5 - x_5 + 3x_5 = -2x_2 + 4x_5 \]Thus the general solution is:
\[ \begin{aligned} x_1 &= -2x_2 + 4x_5 \\ x_3 &= 2x_5 \\ x_4 &= x_5 \end{aligned} \qquad \text{with } x_2, x_5 \in \mathbb{R}. \]The free variables \( x_2 \) and \( x_5 \) can take any real values — hence their name. This parametric form explicitly describes the entire solution set.
For each augmented matrix in row echelon form below, determine which variables are basic and which are free. Assume the variables are \( x_1, x_2, \ldots \) in order.
Matrix 1:
\[ \left[ \begin{array}{ccccc|c} 1 & 4 & 3 & 0 & 0 & 0 \\ 0 & 0 & 1 & -3 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right] \]Matrix 2:
\[ \left[ \begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array} \right] \]Matrix 3:
\[ \left[ \begin{array}{ccccc|c} 1 & 2 & 0 & -1 & -2 & 0 \\ 0 & 1 & 1 & 0 & 3 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right] \]Matrix 4:
\[ \left[ \begin{array}{ccc|c} 1 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right] \]Since these are augmented matrices, the number of variables equals (number of columns − 1). For example, a 5-column augmented matrix has \( 5 - 1 = 4 \) variables: \( x_1, x_2, x_3, x_4 \).