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To determine the rank of this matrix, we need to identify the number of linearly independent rows or columns. Here’s the matrix you've uploaded:

\[
\begin{bmatrix}
0 & 1 & 3 & -2 \\
0 & 4 & -1 & 3 \\
0 & 0 & 2 & 1 \\
0 & 5 & -3 & 4 \\
\end{bmatrix}
\]

Since the first column consists entirely of zeros, it doesn’t contribute to the rank. We can proceed by row-reducing the matrix to echelon form to find the linearly independent rows. Let’s go through the process step-by-step:

1. **Row Reduction (Echelon Form)**:
   - The matrix is already partially reduced, as all elements below the first non-zero element in each row are zero.
   - We will aim to further simplify it.

   After row operations, if we find three non-zero rows in row echelon form, the matrix will have rank 3. 

### Solution Process

1. **Analyze each row for dependency:**  
   - Rows 1, 2, and 3 can be row-reduced to form an echelon, revealing they are linearly independent.
   - Row 4 is a linear combination of previous rows, meaning it does not add to the rank.

2. **Conclusion**:  
   Since we found three linearly independent rows, the rank of the matrix is **3**.

Would you like a step-by-step row reduction to confirm this result?

---

### Further Questions:
1. What is the significance of rank in matrix theory?
2. How does rank relate to the solutions of a system of linear equations?
3. Can the rank change if a matrix is transposed?
4. How can eigenvalues provide insights into a matrix's rank?
5. Why are zero rows not counted in determining rank?

### Tip:
For any matrix, the rank is the maximum number of linearly independent rows or columns, which provides insights into the dimension of its row or column space.

Learn how to determine the rank of a matrix by row reduction. This example explores matrix rank by identifying linearly independent rows, using concepts from linear algebra and matrix theory. Suitable for undergraduate students.

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