Section 2.2 Three elementary row operations
Recall that there are three elementary row operations. These row operations are reversible.
- Add a multiple of a row to another row.
- Multiply a row by a nonzero constant.
- Interchange two rows.
Example:
1. Exchange the first and second rows of the matrix \(A\) below.
\begin{equation*}
A=\begin{pmatrix}
0 \amp 2 \amp -8 \amp 8 \\
1 \amp -2 \amp 1 \amp 0 \\
5 \amp 0 \amp -5 \amp 10
\end{pmatrix}
\end{equation*}
2. Add -5 times of row 1 to row 3.
Exercise: Add -5 times of row 2 to row 3.
3. Multiply \(\frac{1}{30}\) to row 3.
Exercise: Multiply \(\frac{1}{2}\) to row 2.
The matrix \(A\) is called in echelon form.
Exercise: Find the echelon form of the matrix \(B\) below by the methods of elementary row operations, which the leading entries are all one.
\begin{equation*}
B=\begin{pmatrix}
0 \amp 3 \amp -6 \amp 6 \amp 4 \amp -5 \\
3 \amp -7 \amp 8 \amp -5 \amp 8 \amp 9 \\
3 \amp -9 \amp 12 \amp -9 \amp 6 \amp 15
\end{pmatrix}
\end{equation*}
Exercise: Find the reduced row echelon form of the matrices \(A\) and \(B\text{,}\) respectively.
