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}