Question

The value of $a,b,c$ when $\left[\begin{array}{ccc}0& 2b& c\\ a& b& -c\\ a& -b& c\end{array}\right]$ is orthogonal are:

• A. $\pm \frac{1}{\sqrt{2}},\pm \frac{1}{\sqrt{6}},\pm \frac{1}{\sqrt{2}}$
• B. $\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{2}},\pm \frac{1}{\sqrt{6}}$
• C. $\pm \frac{1}{\sqrt{2}},\pm \frac{1}{\sqrt{6}},\pm \frac{1}{\sqrt{3}}$
• D. $\pm \frac{1}{\sqrt{2}},\pm \frac{1}{\sqrt{2}},\pm \frac{1}{\sqrt{2}}$

Answer 1

The correct option is C $\pm \frac{1}{\sqrt{2}},\pm \frac{1}{\sqrt{6}},\pm \frac{1}{\sqrt{3}}$

Given, matrix is orthogonal

$\therefore A{A}^T=I$

$\Rightarrow \left[\begin{array}{ccc}0& 2b& c\\ a& b& -c\\ a& -b& c\end{array}\right]\left[\begin{array}{ccc}0& a& a\\ 2b& b& -b\\ c& -c& c\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\Rightarrow \left[\begin{array}{ccc}4b^2+c^2& 2b^2-c^2& c^2-2b^2\\ 2b^2-c^2& a^2+b^2+c^2& a^2-b^2-c^2\\ c^2+2b^2& a^2-b^2-c^2& a^2+b^2+c^2\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

$\therefore a=\pm \frac{1}{\sqrt{2}},b=\pm \frac{1}{\sqrt{6}},c=\pm \frac{1}{\sqrt{3}}$