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To determine the rank of the matrix \( A \), we will reduce it to its row echelon form and count the number of non-zero rows.

The matrix \( A \) is:

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

### Steps to find the rank:
1. **Transform to Row Echelon Form**:
   We'll apply row operations to get zeros below the leading entries in each row.

   - **Row 1**: \( R_1 = [0, 1, 2] \)
   - **Row 2**: \( R_2 = [2, -1, 0] \)
   - **Row 3**: \( R_3 = [3, 2, 2] \)

2. **Swap Rows** (since the first element in \( R_1 \) is zero):
   Swap \( R_1 \) with \( R_2 \) to get a leading 2 in the first column:
   \[
   A = \begin{bmatrix} 2 & -1 & 0 \\ 0 & 1 & 2 \\ 3 & 2 & 2 \end{bmatrix}
   \]

3. **Create a Zero Below Leading 2 in Row 1**:
   - Make the entry below \( 2 \) in \( R_3 \) zero by using the operation \( R_3 = R_3 - \frac{3}{2}R_1 \):
   \[
   R_3 = [3, 2, 2] - \frac{3}{2}[2, -1, 0] = [0, \frac{7}{2}, 2]
   \]
   The updated matrix becomes:
   \[
   A = \begin{bmatrix} 2 & -1 & 0 \\ 0 & 1 & 2 \\ 0 & \frac{7}{2} & 2 \end{bmatrix}
   \]

4. **Normalize Rows**:
   - Scale \( R_1 \) by \( \frac{1}{2} \) and \( R_3 \) by \( \frac{2}{7} \):
   \[
   A = \begin{bmatrix}

Find the rank of the matrix \( A = \begin{bmatrix} 0 & 1 & 2 \\ 2 & -1 & 0 \\ 3 & 2 & 2 \end{bmatrix} \).

Learn how to determine the rank of the 3x3 matrix \( A = \begin{bmatrix} 0 & 1 & 2 \\ 2 & -1 & 0 \\ 3 & 2 & 2 \end{bmatrix} \) by reducing it to row echelon form. This solution covers fundamental concepts in linear algebra, including matrix rank and row operations, suitable for undergraduate students.

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