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The image shows a matrix problem where you are asked to find the rank of the matrix by reducing it to canonical form (Row Echelon Form). The matrix \( A \) is:

\[
A = \begin{pmatrix}
2 & 1 & 3 & 4 \\
0 & 3 & 4 & 1 \\
2 & 3 & 7 & 5 \\
2 & 5 & 1 & 6
\end{pmatrix}
\]

The steps outlined in the image show how you proceed to reduce the matrix row by row using elementary row operations. Here's a breakdown:

1. **Row 1 (R1):** Dividing \( R1 \) by 2 to make the leading element in the first row a 1.
2. **Row operations on R2, R3, and R4:** After adjusting the first row, you perform row operations to eliminate the first column elements below the pivot in \( R1 \). You perform the following:
   - \( R2 \rightarrow R2 - 2R1 \)
   - \( R3 \rightarrow R3 - 2R1 \)
   - \( R4 \rightarrow R4 - 2R1 \)

3. **Row 2 (R2):** Dividing \( R2 \) by 3 to make the leading coefficient of the second row a 1.
4. **Further row operations:** Eliminating the second column elements in the rows below the pivot in \( R2 \).

5. **Further row reductions** continue until you reach the reduced row echelon form.

The resulting matrix is:

\[
\begin{pmatrix}
1 & 0.5 & 1.5 & 2 \\
0 & 1 & \frac{4}{3} & \frac{1}{3} \\
0 & 0 & 2 & 1 \\
0 & 0 & 4 & 8
\end{pmatrix}
\]

From here, it's clear that the matrix can be further simplified, but it already shows three independent rows. Therefore, **the rank of the matrix is 3** as indicated.

Would you like a detailed explanation of the remaining row operations or additional questions?

Here are five related questions you could explore:
1. What is the formal definition of matrix rank?
2. How does the number of leading 1's in row echelon form relate to matrix rank?
3. Can rank be used to determine whether a system of equations has unique, infinite, or no solutions?
4. How do we handle matrices with more columns than rows when calculating the rank?
5. What is the difference between row echelon form and reduced row echelon form?

**Tip:** The rank of a matrix gives the number of linearly independent rows (or columns).

Find the rank of the matrix by using canonical form.

This page walks you through finding the rank of a matrix by using canonical (row echelon) form. The process includes performing elementary row operations to reduce the matrix to its echelon form and determining the rank based on non-zero rows. Suitable for undergraduate or advanced high school students studying linear algebra.

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