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The problem asks you to perform row operations to reduce the matrix to its row-reduced form. Let's go step by step and solve this.

The matrix given is:

\[
\begin{pmatrix}
1 & 2 & -2 & -1 \\
0 & 3 & -15 & 1 \\
0 & -1 & 5 & -\frac{1}{3}
\end{pmatrix}
\]

### Step 1: The goal is to get zeros below and above the pivots and reduce the matrix to its simplest form.

- The pivot in the first column is already \(1\) in row 1, so we don’t need to change row 1.

### Step 2: Eliminate the non-zero entry in column 2 below the pivot.

- To eliminate the \( -1 \) in row 3, column 2, we can add \( \frac{1}{3} \) times row 2 to row 3:
  \[
  R_3 = R_3 + \frac{1}{3}R_2
  \]
  Performing this gives the updated matrix:

\[
\begin{pmatrix}
1 & 2 & -2 & -1 \\
0 & 3 & -15 & 1 \\
0 & 0 & 0 & 0
\end{pmatrix}
\]

This is the reduced row echelon form (RREF).

Would you like further details on any specific part of the process?

Here are 5 follow-up questions:
1. What are the different types of elementary row operations?
2. How is the row-reduced form of a matrix used to solve systems of linear equations?
3. What are the conditions for a matrix to be in row-reduced form?
4. Can every matrix be reduced to row-reduced form?
5. How does the number of pivots in the reduced matrix relate to the solution of a system?

**Tip:** Always aim to use the smallest row operations to simplify computations when performing Gaussian elimination.

Use row operations to change the matrix below to reduced form: [ 1 2 -2 -1 ; 0 3 -15 1 ; 0 -1 5 -1/3 ].

This problem demonstrates the steps to perform elementary row operations to reduce a 3x4 matrix to its row-reduced echelon form (RREF). Learn how Gaussian elimination is applied to matrices in this detailed step-by-step solution, suitable for students in Grades 10-12 or early college.

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