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It appears that you're working on determining the rank of a matrix using its canonical form. Let's walk through the solution steps presented in the image:

1. **Matrix \( A \)**:
   The matrix you're working with is:
   \[
   A = \begin{bmatrix}
   2 & 1 & 3 & 4 \\
   0 & 3 & 4 & 1 \\
   2 & 3 & 7 & 5 \\
   2 & 5 & 11 & 6
   \end{bmatrix}
   \]

2. **Row Reduction**:
   The steps show a process of performing row operations to transform the matrix into row echelon form.

3. **Row Operations**:
   - First, \( R_1 \) is divided by 2.
   - Then \( R_3 \) is replaced by \( R_3 - 2R_1 \).
   - \( R_4 \) is replaced by \( R_4 - 2R_1 \).
   - Further row reductions are performed to simplify the matrix.

4. **Final Matrix**:
   After performing all the row operations, the matrix becomes:
   \[
   \begin{bmatrix}
   1 & 0.5 & 1.5 & 2 \\
   0 & 2 & 4 & 1 \\
   0 & 0 & 4 & 1 \\
   0 & 0 & 4 & 8 & 2
   \end{bmatrix}
   \]
   From this, you can see that there are 3 non-zero rows, indicating that the rank of the matrix is **3**.

The conclusion seems correct based on the row-reduction process, and the matrix's rank is indeed **3**.

Let me know if you want further details on the row operations or need help with a related topic.

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Here are 5 related questions to deepen your understanding:

1. What is the definition of the rank of a matrix?
2. Why is the canonical form useful for finding the rank of a matrix?
3. Can a matrix have a rank larger than the number of rows or columns?
4. How can you determine the rank of a matrix using determinants?
5. What is the relationship between the rank of a matrix and its nullity?

**Tip**: When performing row operations, always aim to simplify the leading entries of each row (pivot points) while keeping the rows above and below them zero.

Find the rank of the matrix using canonical form.

Learn how to determine the rank of a matrix using canonical form. The problem includes step-by-step row reductions and operations to transform the matrix into row echelon form. Ideal for students in Grades 11-12 or college level.

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