The matrix `A=1/3{:[(1,2,2),(2,1,-2),(-2,2,-1)]:}` is 1) orthogonal 2) involutory 3) idempotent 4) nilpotent

Options

A. orthogonal
B. involutory
C. idempotent
D. nilpotent

Explanation

To determine the properties of the matrix \( A = \frac{1}{3} \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ -2 & 2 & -1 \end{pmatrix} \), we will check if it is orthogonal, involutory, idempotent, or nilpotent.

### Step 1: Check if \( A \) is Orthogonal
A matrix \( A \) is orthogonal if \( A A^T = I \), where \( A^T \) is the transpose of \( A \) and \( I \) is the identity matrix.

1. **Find the transpose \( A^T \)**:
\[ A^T = \frac{1}{3} \begin{pmatrix} 1 & 2 & -2 \\ 2 & 1 & 2 \\ 2 & -2 & -1 \end{pmatrix} \]

2. **Calculate \( A A^T \)**:
\[ A A^T = \frac{1}{3} \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ -2 & 2 & -1 \end{pmatrix} \cdot \frac{1}{3} \begin{pmatrix} 1 & 2 & -2 \\ 2 & 1 & 2 \\ 2 & -2 & -1 \end{pmatrix} \]
- Calculate each element of the resulting matrix.

3. **Result of \( A A^T \)**:
After performing the multiplication, we find:
\[ A A^T = \frac{1}{9} \begin{pmatrix} 9 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 9 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} = I \]
Thus, \( A \) is orthogonal.

### Step 2: Check if \( A \) is Involutory
A matrix \( A \) is involutory if \( A^2 = I \).

1. **Calculate \( A^2 \)**:
\[ A^2 = A \cdot A = \frac{1}{3} \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ -2 & 2 & -1 \end{pmatrix} \cdot \frac{1}{3} \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ -2 & 2 & -1 \end{pmatrix} \]
- Perform the multiplication and check if it equals \( I \).

2. **Result of \( A^2 \)**:
After performing the multiplication, we find that \( A^2 \neq I \). Therefore, \( A \) is not involutory.

### Step 3: Check if \( A \) is Idempotent
A matrix \( A \) is idempotent if \( A^2 = A \).

1. **Using the previous result**:
Since we already calculated \( A^2 \) and found it does not equal \( A \), we conclude that \( A \) is not idempotent.

### Step 4: Check if \( A \) is Nilpotent
A matrix \( A \) is nilpotent if \( A^m = 0 \) for some positive integer \( m \).

1. **Using the previous result**:
Since \( A^2 \neq 0 \) and \( A \) is not a zero matrix, \( A \) is not nilpotent.

### Conclusion
The only property that holds for the matrix \( A \) is that it is orthogonal.

### Final Answer
1) Orthogonal