Ge Example By Pierce

Find the reduced row echelon form of the matrix:

(1)
\begin{align} \left(\begin{array}{cc} 2 & 2 \\ 1 & 1 \\ -1 & -2 \\ 3 & 3 \end{array}\right) \end{align}

- Virgil Pierce

1. We begin by identifying a pivot in the first column. We can use any of the nonzero entries. Take the 2.
2. Divide the first row by 2.

(2)
\begin{align} \left(\begin{array}{cc} 1 & 1 \\ 1 & 1 \\ -1 & -2 \\ 3 & 3 \end{array}\right) \end{align}

3. Subtract the first row from the 2nd.

(3)
\begin{align} \left(\begin{array}{cc} 1 & 1 \\ 0 & 0 \\ -1 & -2 \\ 3 & 3 \end{array}\right) \end{align}

4. Add the first row to the 3rd.

(4)
\begin{align} \left(\begin{array}{cc} 1 & 1 \\ 0 & 0 \\ 0 & -1 \\ 3 & 3 \end{array}\right) \end{align}

5. Subtract 3 times the first row from the 4th.

(5)
\begin{align} \left(\begin{array}{cc} 1 & 1 \\ 0 & 0 \\ 0 & -1 \\ 0 & 0 \end{array}\right) \end{align}

6. We are done with the first column. Identify a pivot in the second column, note that only the -1 will work.
7. Permute the 2nd and 3rd rows.

(6)
\begin{align} \left(\begin{array}{cc} 1 & 1 \\ 0 & -1 \\ 0 & 0 \\ 0 & 0 \end{array}\right) \end{align}

8. Divide the 2nd row by -1.

(7)
\begin{align} \left(\begin{array}{cc} 1 & 1 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{array}\right) \end{align}
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