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To calculate the rank of the given matrix \( A \), we need to determine the maximum number of linearly independent rows or columns. The matrix \( A \) is:

\[
A = \begin{pmatrix}
3 & -1 & 1 \\
2 & 1 & 0 \\
1 & 2 & -1
\end{pmatrix}
\]

### Step 1: Convert the matrix to Row Echelon Form (REF)
We will use elementary row operations to bring the matrix to row echelon form.

\[
\text{Starting matrix:} \quad \begin{pmatrix} 3 & -1 & 1 \\ 2 & 1 & 0 \\ 1 & 2 & -1 \end{pmatrix}
\]

1. **Row 2 operation**: Replace \( R_2 \) with \( R_2 - \frac{2}{3}R_1 \):
   \[
   R_2 \rightarrow R_2 - \frac{2}{3}R_1
   \]
   \[
   \begin{pmatrix}
   3 & -1 & 1 \\
   0 & \frac{5}{3} & -\frac{2}{3} \\
   1 & 2 & -1
   \end{pmatrix}
   \]

2. **Row 3 operation**: Replace \( R_3 \) with \( R_3 - \frac{1}{3}R_1 \):
   \[
   R_3 \rightarrow R_3 - \frac{1}{3}R_1
   \]
   \[
   \begin{pmatrix}
   3 & -1 & 1 \\
   0 & \frac{5}{3} & -\frac{2}{3} \\
   0 & \frac{7}{3} & -\frac{4}{3}
   \end{pmatrix}
   \]

3. **Row 3 operation**: Replace \( R_3 \) with \( R_3 - \frac{7}{5}R_2 \):
   \[
   R_3 \rightarrow R_3 - \frac{7}{5}R_2
   \]
   \[
   \begin{pmatrix}
   3 & -1 & 1 \\
   0 & \frac{5}{3} & -\frac{2}{3} \\
   0 & 0 & 0
   \end{pmatrix}
   \]

### Step 2: Determine the Rank
The row echelon form of the matrix has two non-zero rows. Therefore, the rank of the matrix \( A \) is **2**.

Would you like any further details or have any questions? Here are some related questions you might find helpful:

1. What are the definitions and differences between rank and nullity of a matrix?
2. How do elementary row operations affect the determinant and rank of a matrix?
3. Can the rank of a matrix change if the matrix is multiplied by a scalar?
4. How can the rank be determined using the column echelon form?
5. How is the rank related to the solutions of a system of linear equations?
6. What is the significance of a full-rank matrix in linear algebra?
7. How do you determine if two matrices are equivalent in terms of rank?
8. What are some applications of matrix rank in machine learning and data science?

**Tip:** The rank of a matrix is always less than or equal to the smallest dimension of the matrix (i.e., min(number of rows, number of columns)).

Learn how to calculate the rank of matrix A using row echelon form with a detailed step-by-step solution. This problem covers matrix operations, linear independence, and row echelon form. Suitable for advanced high school students and college-level learners.

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