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Calculate the rank of matrix

A= 3  -1 1
2   1. 0
1. 2.  -1

To calculate the rank of the matrix \( A = \begin{pmatrix} 3 & -1 & 1 \\ 2 & 1 & 0 \\ 1 & 2 & -1 \end{pmatrix} \), we need to perform row reduction to bring the matrix to its row echelon form (REF) or reduced row echelon form (RREF). The rank is the number of non-zero rows in this form.

Let's proceed with the row reduction:

1. Start with the matrix \( A \):
   \[
   A = \begin{pmatrix} 
   3 & -1 & 1 \\
   2 & 1 & 0 \\
   1 & 2 & -1 
   \end{pmatrix}
   \]

2. **Row 2**: Subtract \(\frac{2}{3}\) times **Row 1** from **Row 2**:
   \[
   R2 \rightarrow R2 - \frac{2}{3}R1
   \]
   \[
   \begin{pmatrix} 
   3 & -1 & 1 \\
   0 & \frac{5}{3} & -\frac{2}{3} \\
   1 & 2 & -1 
   \end{pmatrix}
   \]

3. **Row 3**: Subtract \(\frac{1}{3}\) times **Row 1** from **Row 3**:
   \[
   R3 \rightarrow R3 - \frac{1}{3}R1
   \]
   \[
   \begin{pmatrix} 
   3 & -1 & 1 \\
   0 & \frac{5}{3} & -\frac{2}{3} \\
   0 & \frac{7}{3} & -\frac{4}{3} 
   \end{pmatrix}
   \]

4. **Row 3**: Subtract \(\frac{7}{5}\) times **Row 2** from **Row 3**:
   \[
   R3 \rightarrow R3 - \frac{7}{5}R2
   \]
   \[
   \begin{pmatrix} 
   3 & -1 & 1 \\
   0 & \frac{5}{3} & -\frac{2}{3} \\
   0 & 0 & 0 
   \end{pmatrix}
   \]

In this row echelon form, we have two non-zero rows. Therefore, the rank of the matrix is 2.

### Summary:
- **Rank of the matrix \( A \) is 2**.

Would you like more details on any specific step, or do you have any other questions?

Here are some related questions you might find interesting:
1. How is row echelon form different from reduced row echelon form?
2. What is the significance of a matrix's rank in solving linear systems?
3. How do you determine if a matrix is invertible using its rank?
4. Can a matrix have a rank greater than its number of rows or columns?
5. What is the relationship between the rank of a matrix and its nullity?
6. How does the rank of a matrix relate to the solutions of a system of linear equations?
7. What is the maximum possible rank of a 3x3 matrix?
8. How would the rank change if one row of the matrix was a multiple of another?

**Tip**: The rank of a matrix is equal to the maximum number of linearly independent row or column vectors in the matrix.

Learn how to calculate the rank of matrix A = [3 -1 1; 2 1 0; 1 2 -1] using row reduction to row echelon form (REF). Understand the significance of matrix rank in linear algebra. Suitable for college-level students and enthusiasts of linear algebra concepts.

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