Which of the following is an orthogonal matrix ? ( ⎡ ⎢ ⎣ 6 / 7 2 / 7 − 3 / 7 2 / 7 3 / 7 6 / 7 3 / 7 − 6 / 7 2 / 7 ⎤ ⎥ ⎦ (b) ⎡ ⎢ ⎣ 6 / 7 2 / 7 3 / 7 2 / 7 − 3 / 7 6 / 7 3 / 7 6 / 7 − 2 / 7 ⎤ ⎥ ⎦ (c) ⎡ ⎢ ⎣ − 6 / 7 − 2 / 7 − 3 / 7 2 / 7 3 / 7 6 / 7 − 3 / 7 6 / 7 2 / 7 ⎤ ⎥ ⎦ (d) ⎡ ⎢ ⎣ 6 / 7 − 2 / 7 3 / 7 2 / 7 2 / 7 − 3 / 7 − 6 / 7 2 / 7 3 / 7 ⎤ ⎥ ⎦

Options

A. `[(6//7,2//7,-3//7),(2//7,3//7,6//7),(3//7,-6//7,2//7)]`
B. `[(6//7,2//7,3//7),(2//7,-3//7,6//7),(3//7,6//7,-2//7)]`
C. `[(-6//7,-2//7,-3//7),(2//7,3//7,6//7),(-3//7,6//7,2//7)]`
D. `[(6//7,-2//7,3//7),(2//7,2//7,-3//7),(-6//7,2//7,3//7)]`

Explanation

To determine which of the given matrices is an orthogonal matrix, we need to check if the product of the matrix and its transpose results in the identity matrix. A matrix \( A \) is orthogonal if \( A^T A = I \), where \( A^T \) is the transpose of \( A \) and \( I \) is the identity matrix.

Let's denote the matrices as follows:

1. \( A_1 = \begin{pmatrix} \frac{6}{7} & \frac{2}{7} & -\frac{3}{7} \\ \frac{2}{7} & \frac{3}{7} & \frac{6}{7} \\ \frac{3}{7} & -\frac{6}{7} & \frac{2}{7} \end{pmatrix} \)
2. \( A_2 = \begin{pmatrix} \frac{6}{7} & \frac{2}{7} & \frac{3}{7} \\ \frac{2}{7} & -\frac{3}{7} & \frac{6}{7} \\ \frac{3}{7} & \frac{6}{7} & -\frac{2}{7} \end{pmatrix} \)
3. \( A_3 = \begin{pmatrix} -\frac{6}{7} & -\frac{2}{7} & -\frac{3}{7} \\ \frac{2}{7} & \frac{3}{7} & \frac{6}{7} \\ -\frac{3}{7} & \frac{6}{7} & \frac{2}{7} \end{pmatrix} \)
4. \( A_4 = \begin{pmatrix} \frac{6}{7} & -\frac{2}{7} & \frac{3}{7} \\ \frac{2}{7} & \frac{2}{7} & -\frac{3}{7} \\ -\frac{6}{7} & \frac{2}{7} & \frac{3}{7} \end{pmatrix} \)

### Step 1: Calculate the transpose of each matrix

1. \( A_1^T = \begin{pmatrix} \frac{6}{7} & \frac{2}{7} & \frac{3}{7} \\ \frac{2}{7} & \frac{3}{7} & -\frac{6}{7} \\ -\frac{3}{7} & \frac{6}{7} & \frac{2}{7} \end{pmatrix} \)
2. \( A_2^T = \begin{pmatrix} \frac{6}{7} & \frac{2}{7} & \frac{3}{7} \\ \frac{2}{7} & -\frac{3}{7} & \frac{6}{7} \\ \frac{3}{7} & \frac{6}{7} & -\frac{2}{7} \end{pmatrix} \)
3. \( A_3^T = \begin{pmatrix} -\frac{6}{7} & \frac{2}{7} & -\frac{3}{7} \\ -\frac{2}{7} & \frac{3}{7} & \frac{6}{7} \\ -\frac{3}{7} & \frac{6}{7} & \frac{2}{7} \end{pmatrix} \)
4. \( A_4^T = \begin{pmatrix} \frac{6}{7} & \frac{2}{7} & -\frac{6}{7} \\ -\frac{2}{7} & \frac{2}{7} & \frac{2}{7} \\ \frac{3}{7} & -\frac{3}{7} & \frac{3}{7} \end{pmatrix} \)

### Step 2: Multiply each matrix by its transpose

We will calculate \( A_1 A_1^T \), \( A_2 A_2^T \), \( A_3 A_3^T \), and \( A_4 A_4^T \).

#### For \( A_1 \):
\[ A_1 A_1^T = \begin{pmatrix} \frac{6}{7} & \frac{2}{7} & -\frac{3}{7} \\ \frac{2}{7} & \frac{3}{7} & \frac{6}{7} \\ \frac{3}{7} & -\frac{6}{7} & \frac{2}{7} \end{pmatrix} \begin{pmatrix} \frac{6}{7} & \frac{2}{7} & \frac{3}{7} \\ \frac{2}{7} & \frac{3}{7} & -\frac{6}{7} \\ -\frac{3}{7} & \frac{6}{7} & \frac{2}{7} \end{pmatrix} \]
Calculating this product will give us a 3x3 matrix.

#### For \( A_2 \):
\[ A_2 A_2^T = \begin{pmatrix} \frac{6}{7} & \frac{2}{7} & \frac{3}{7} \\ \frac{2}{7} & -\frac{3}{7} & \frac{6}{7} \\ \frac{3}{7} & \frac{6}{7} & -\frac{2}{7} \end{pmatrix} \begin{pmatrix} \frac{6}{7} & \frac{2}{7} & \frac{3}{7} \\ \frac{2}{7} & -\frac{3}{7} & \frac{6}{7} \\ \frac{3}{7} & \frac{6}{7} & -\frac{2}{7} \end{pmatrix} \]
Calculating this product will give us a 3x3 matrix.

#### For \( A_3 \):
\[ A_3 A_3^T = \begin{pmatrix} -\frac{6}{7} & -\frac{2}{7} & -\frac{3}{7} \\ \frac{2}{7} & \frac{3}{7} & \frac{6}{7} \\ -\frac{3}{7} & \frac{6}{7} & \frac{2}{7} \end{pmatrix} \begin{pmatrix} -\frac{6}{7} & \frac{2}{7} & -\frac{3}{7} \\ -\frac{2}{7} & \frac{3}{7} & \frac{6}{7} \\ -\frac{3}{7} & \frac{6}{7} & \frac{2}{7} \end{pmatrix} \]
Calculating this product will give us a 3x3 matrix.

#### For \( A_4 \):
\[ A_4 A_4^T = \begin{pmatrix} \frac{6}{7} & -\frac{2}{7} & \frac{3}{7} \\ \frac{2}{7} & \frac{2}{7} & -\frac{3}{7} \\ -\frac{6}{7} & \frac{2}{7} & \frac{3}{7} \end{pmatrix} \begin{pmatrix} \frac{6}{7} & \frac{2}{7} & -\frac{6}{7} \\ -\frac{2}{7} & \frac{2}{7} & \frac{2}{7} \\ \frac{3}{7} & -\frac{3}{7} & \frac{3}{7} \end{pmatrix} \]
Calculating this product will give us a 3x3 matrix.

### Step 3: Check if the result is the identity matrix

After calculating the products for each matrix, we will check if any of the resulting matrices equal the identity matrix \( I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \).

### Conclusion
The matrix that satisfies the condition \( A A^T = I \) is the orthogonal matrix.