Question

If A =$\begin{bmatrix} 0 & \alpha & \alpha \\ 2\beta & \beta & - \beta \\ \gamma & - \gamma & \gamma \end{bmatrix}$ is an orthogonal matrix, then the

number of possible triplets (a, b, g)

• A. 8
• B. 6
• C. 4
• D. 2

Answer 1

Answer: (A)

8

Answer 2

AA^T = I

$\begin{bmatrix} 0 & \alpha & \alpha \\ 2\beta & \beta & - \beta \\ \gamma & - \gamma & \gamma \end{bmatrix}$

$\begin{bmatrix} 0 & 2\beta & \gamma \\ \alpha & \beta & - \gamma \\ \alpha & - \beta & \gamma \end{bmatrix}$=$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

̃ $\begin{bmatrix} 2\alpha^{2} & 0 & 0 \\ 0 & 6\beta^{2} & 0 \\ 0 & 0 & 3\gamma^{2} \end{bmatrix}$ $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

̃ 2a² = 1, 6b² = 1, 3g² = 1

\ a = ± $\frac{1}{\sqrt{2}}$, b = ± $\frac{1}{\sqrt{6}}$, g = ± $\frac{1}{\sqrt{3}}$