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To determine whether the columns of matrix \( A \) are linearly dependent or independent, we need to perform Gaussian elimination (row reduction). Let's break this down step by step:

The given matrix is:
\[
A = \begin{pmatrix}
-5 & -25 & 20 \\
1 & 2 & -7 \\
3 & 14 & -13
\end{pmatrix}
\]

### Step 1: Eliminate the first entry in the second and third rows.

We want to make the elements below the pivot in the first column zero.

- To eliminate the \( 1 \) in the second row, we add \( \frac{1}{-5} = -\frac{1}{5} \) times the first row to the second row.
  
  New second row:
  \[
  R_2 = R_2 + \left( -\frac{1}{5} \right) R_1
  \]
  Calculation:
  \[
  \begin{aligned}
  1 + \left( -\frac{1}{5} \right)(-5) &= 0 \\
  2 + \left( -\frac{1}{5} \right)(-25) &= 7 \\
  -7 + \left( -\frac{1}{5} \right)(20) &= -11
  \end{aligned}
  \]
  So the new second row is \( \begin{pmatrix} 0 & 7 & -11 \end{pmatrix} \).

- To eliminate the \( 3 \) in the third row, we add \( \frac{3}{-5} = -\frac{3}{5} \) times the first row to the third row.

  New third row:
  \[
  R_3 = R_3 + \left( -\frac{3}{5} \right) R_1
  \]
  Calculation:
  \[
  \begin{aligned}
  3 + \left( -\frac{3}{5} \right)(-5) &= 0 \\
  14 + \left( -\frac{3}{5} \right)(-25) &= -1 \\
  -13 + \left( -\frac{3}{5} \right)(20) &= -25
  \end{aligned}
  \]
  So the new third row is \( \begin{pmatrix} 0 & -1 & -25 \end{pmatrix} \).

At this point, the matrix is:
\[
A = \begin{pmatrix}
-5 & -25 & 20 \\
0 & 7 & -11 \\
0 & -1 & -25
\end{pmatrix}
\]

### Step 2: Eliminate the second entry in the third row.

Now, to eliminate the \( -1 \) in the third row, we add \( \frac{-1}{7} = \frac{1}{7} \) times the second row to the third row.

New third row:
\[
R_3 = R_3 + \left( \frac{1}{7} \right) R_2
\]
Calculation:
\[
\begin{aligned}
-1 + \left( \frac{1}{7} \right)(7) &= 0 \\
-25 + \left( \frac{1}{7} \right)(-11) &= -25 - \frac{11}{7}
\end{aligned}
\]
The third row becomes \( \begin{pmatrix} 0 & 0 & -\frac{186}{7} \end{pmatrix} \).

### Step 3: Conclusion

The matrix is now in row echelon form:
\[
\begin{pmatrix}
-5 & -25 & 20 \\
0 & 7 & -11 \\
0 & 0 & -\frac{186}{7}
\end{pmatrix}
\]
Since the matrix has a pivot in every column, the rank of the matrix is 3, which means that the columns are linearly dependent.

Thus, the answer is:
- **A. The columns of \( A \) are linearly dependent.**

---

Would you like any further details or clarifications?

Here are some related questions for deeper understanding:
1. What is the significance of a pivot column in Gaussian elimination?
2. How do we define linear dependence and independence in the context of column vectors?
3. How would we determine the rank of a matrix if it has dependent columns?
4. Can a square matrix have linearly independent columns but a determinant of zero?
5. How does Gaussian elimination differ from Gauss-Jordan elimination?

**Tip:** In row reduction, always focus on eliminating entries systematically to get a triangular matrix, which simplifies determining dependence.

Let A be a 3x3 matrix given by A = [[-5, -25, 20], [1, 2, -7], [3, 14, -13]]. We want to determine if the columns of matrix A are linearly independent. To do this, we perform row reduction on A as described.

Learn how to determine the linear dependence of matrix columns using Gaussian elimination. This example uses a 3x3 matrix and involves detailed row reduction steps to analyze linear independence. Suitable for college-level students studying Linear Algebra.

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