Question

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

• A. a = ±$\frac{1}{\sqrt{2}}$
• B. b = ±$\frac{1}{\sqrt{6}}$
• C. g = ± $\frac{1}{\sqrt{3}}$
• D. all of these

Answer 1

Answer: (D)

all of these

Answer 2

Let A = $\begin{bmatrix} 0 & 2\beta & \gamma \\ \alpha & \beta & - \gamma \\ \alpha & - \beta & \gamma \end{bmatrix}$, A¢ = $\begin{bmatrix} 0 & \alpha & \alpha \\ 2\beta & \beta & - \beta \\ \gamma & - \gamma & \gamma \end{bmatrix}$

Since A is orthogonal, \ AA¢ = I

Ž$\left[ \begin{array} { c c c } 0 & 2 \beta & \gamma \\ \alpha & \beta & - \gamma \\ \alpha & - \beta & \gamma \end{array} \right]$ $\begin{bmatrix} 0 & \alpha & \alpha \\ 2\beta & \beta & - \beta \\ \gamma & - \gamma & \gamma \end{bmatrix}$=$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

Ž $\begin{bmatrix} 4\beta^{2} + \gamma^{2} & 2\beta^{2} - \gamma^{2} & - 2\beta^{2} + \gamma^{2} \ 2\beta^{2} - \gamma^{2} & \alpha^{2} + \beta^{2} + \gamma^{2} & \alpha^{2} - \beta^{2} - \gamma^{2} \

• 2\beta^{2} + \gamma^{2} & \alpha^{2} - \beta^{2} - \gamma^{2} & \alpha^{2} + \beta^{2} + \gamma^{2} \end{bmatrix}$

= $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

Equation the corresponding elements, we have

$\left. \ \begin{matrix} 4\beta^{2} + \gamma^{2} = 1 \\ 2\beta^{2} - \gamma^{2} = 0 \end{matrix} \right\}$Ž b = ± $\frac{1}{\sqrt{6}}$, g = ± $\frac{1}{\sqrt{3}}$

a² + b² + g² = 1 Ž a² +$\frac{1}{6}$+$\frac{1}{3}$= 1 Ž a = ±$\frac{1}{\sqrt{2}}$.

Hence (4) is correct answer.