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The task requires us to find the **row-echelon form** of the given matrix:

\[
\begin{bmatrix}
2 & 3 & 4 \\
1 & -1 & 2 \\
2 & -5 & 6
\end{bmatrix}
\]

### Step-by-Step Procedure:

1. **Make the first pivot** (top-left) 1. We can do this by dividing the first row by 2.

\[
\begin{bmatrix}
1 & \frac{3}{2} & 2 \\
1 & -1 & 2 \\
2 & -5 & 6
\end{bmatrix}
\]

2. **Eliminate the entries below the pivot**. Subtract Row 1 from Row 2 and Row 1 multiplied by 2 from Row 3.

- \( R_2 = R_2 - R_1 \)
- \( R_3 = R_3 - 2 \cdot R_1 \)

\[
\begin{bmatrix}
1 & \frac{3}{2} & 2 \\
0 & -\frac{5}{2} & 0 \\
0 & -\frac{11}{2} & 2
\end{bmatrix}
\]

3. **Make the second pivot** (second row, second column) a 1. We do this by dividing the second row by \( -\frac{5}{2} \).

\[
\begin{bmatrix}
1 & \frac{3}{2} & 2 \\
0 & 1 & 0 \\
0 & -\frac{11}{2} & 2
\end{bmatrix}
\]

4. **Eliminate the entries below the second pivot**. Subtract \( \frac{11}{2} \times \text{Row 2} \) from Row 3.

\[
R_3 = R_3 + \frac{11}{2} \times R_2
\]

This simplifies the matrix to:

\[
\begin{bmatrix}
1 & \frac{3}{2} & 2 \\
0 & 1 & 0 \\
0 & 0 & 2
\end{bmatrix}
\]

5. **Make the last pivot** (third row, third column) a 1. Divide Row 3 by 2.

\[
\begin{bmatrix}
1 & \frac{3}{2} & 2 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
\]

This is the **row-echelon form** of the matrix.

Would you like further details on any of the steps?

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Here are 5 related questions for deeper understanding:

1. How does row-echelon form help in solving systems of linear equations?
2. What is the difference between row-echelon form and reduced row-echelon form?
3. How can you determine the rank of a matrix from its row-echelon form?
4. How does Gaussian elimination relate to row-echelon form?
5. What role do pivot elements play in transforming a matrix to row-echelon form?

**Tip**: When performing row operations, always track the operations carefully, as even small errors can lead to incorrect results!

Find the row-echelon form of the matrix: [[2, 3, 4], [1, -1, 2], [2, -5, 6]]

Learn how to convert the matrix [[2, 3, 4], [1, -1, 2], [2, -5, 6]] into row-echelon form using Gaussian elimination and elementary row operations. This step-by-step solution is perfect for college-level students studying Linear Algebra.

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