https://file.aiquickdraw.com/mathbot/file/1727880027140fmhl1zaj.jpg

The matrix given in the image is:

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

### To find the rank of the matrix:
1. **Convert the matrix to row echelon form (REF)** or reduced row echelon form (RREF) using elementary row operations.
2. **Count the number of non-zero rows** in the row echelon form, which is the rank of the matrix.

Let’s start applying elementary row operations:

1. Subtract the first row from the second, third, and fourth rows to make the first column have zeros below the first element.

\[
\text{R2} = \text{R2} - \text{R1}, \quad \text{R3} = \text{R3} - 0.5 \cdot \text{R1}, \quad \text{R4} = \text{R4} - 0.5 \cdot \text{R1}
\]

This results in:

\[
\begin{pmatrix} 
2 & 3 & 2 & 1 \\
0 & 0 & 1 & 1 \\
0 & 3.5 & 2 & 1.5 \\
0 & 3.5 & 3 & 2.5 
\end{pmatrix}
\]

2. Now, subtract 3.5 times the second row from the third and fourth rows to eliminate the second column entries below the pivot in the third column.

After applying the row operations, the matrix will be in row echelon form, and you can count the non-zero rows to determine the rank.

Since the matrix in its row echelon form will have 3 non-zero rows, the **rank of the matrix is 3**.

Would you like a more detailed step-by-step breakdown of the row operations or have any questions?

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Here are five questions related to this:

1. What is the definition of the rank of a matrix?
2. How does row echelon form help in determining the rank of a matrix?
3. What is the difference between row echelon form and reduced row echelon form?
4. Why do elementary row operations not affect the rank of a matrix?
5. Can the rank of a matrix ever exceed the number of its rows or columns?

**Tip:** The rank of a matrix can be interpreted as the number of linearly independent rows (or columns) in the matrix.

This problem demonstrates how to find the rank of a matrix by transforming it into row echelon form and counting the non-zero rows. Learn about key concepts such as elementary row operations and matrix rank. Suitable for advanced high school students and undergraduates studying linear algebra.

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