Section 4.2 Pivots
A leading coefficient in a matrix is the first non-zero entry in a given row. The position of a leading coefficient in the reduced row echelon matrix of \(A\) is called a pivot position.
Sage has the
pivot() method to identify the pivot columns of the reduced row echelon form of a given matrix. Here is an example with \(\textbf{Case I}\text{:}\)
Observe that the solution obtained means that there are pivot positions in the first, second and third columns of the reduced matrix. We can easily verify this by revisiting the reduced form of the matrix:
We can clearly see the pivot positions in the first three columns.
Notice that Sage can determine which columns contain the pivot positions in the reduced row echelon form of the matrix without explicitly showing this form.
The pivot positions in the reduced row echelon form of the augmented matrix of a linear system are crucial for determining the nature of its solutions.
Let \(R\) and \(Rb\) denote the reduced row echelon forms of the coefficient matrix and the augmented matrix, respectively.
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A system is inconsistent exactly when a pivot position appears in the last column of \(Rb\text{.}\)
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If the system is consistent, then the columns of \(R\) containing pivot positions correspond to basic variables (dependent), while the columns without pivot positions correspond to free variables (independent).
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The solution is unique when every column of \(R\) contains a pivot position, and there are infinitely many solutions when at least one column of R does not contain a pivot position.
Subsection 4.2.1 Compatible Unique Solutions (Case I)
Observe that in our previous system there was no pivot position in the last column, so the system was compatible. Moreover, every column of the reduced row echelon form of the coefficient matrix had a pivot position, so the system has a unique solution.
Recall that in Section 3.2, this unique solution was explicitly obtained.
This coincides with the pivot criterion, as it implies a unique solution for this matrix.
Subsection 4.2.2 Compatible Infinitely Many Solutions (Case II)
Let’s find the pivot columns for our \(\textbf{Case II}\) system.
We found that there are pivot positions in the first and second columns.
Since there is no pivot position in the last column, the system is consistent. Because there are columns without pivot positions in the rest of the matrix, the system has infinitely many solutions. The first and second columns correspond to the variables \(x\) and \(y\text{,}\) so these are the basic variables, while \(z\) is a free variable. Therefore, the solution can be written as a function of the parameter \(z\text{.}\)
\begin{gather*}
\left\{
\begin{array}{l}
x + 2z = 4\\
y - z = 0\\
0 = 0
\end{array}
\right.
\hspace{6pt}
\Rightarrow
\hspace{6pt}
\left\{
\begin{array}{l}
x = -2z + 4\\
y = z\\
z = z
\end{array}
\right.
\end{gather*}
This coincides with the solutions we found in Section 3.2, namely a parameterized solution, but in this case Sage defaults to
r1 as a parameter.
Subsection 4.2.3 Incompatible (Case III)
Let’s find the pivot columns for our \(\textbf{Case III}\) system.
We found that there are pivot positions in the first, second and fourth columns. In particular, there is a pivot position in the last column so the system is incompatible.
This is again consistent with the solution from Section 3.2, as an empty list of solutions was returned.
