Question

If $A=\frac{1}{3}\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ a& 2& b\end{array}\right]$ is an orthogonal matrix, then

• A. a = - 2, b = - 1
• B. a = 2, b = 1
• C. a = 2, b = - 1
• D. a = - 2, b = 1

Answer 1

Answer: (A)

As A is an orthogonal matrix, $A{A}^T=I$
$\Rightarrow \frac{1}{3}\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ a& 2& b\end{array}\right].\frac{1}{3}\left[\begin{array}{ccc}1& 2& a\\ 2& 1& 2\\ 2& -2& b\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$\Rightarrow \frac{1}{9}\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ a& 2& b\end{array}\right]\left[\begin{array}{ccc}1& 2& a\\ 2& 1& 2\\ 2& -2& b\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$\Rightarrow \left[\begin{array}{ccc}9& 0& a+4+2b\\ 0& 9& 2a+2-2b\\ a+4+2b& 2a+2-2b& a^2+4+b^2\end{array}\right]=\left[\begin{array}{ccc}9& 0& 0\\ 0& 9& 0\\ 0& 0& 9\end{array}\right]$
$\Rightarrow a+4+2b=0,2a+2-2b=0,a^2+4+b^2=9$
$\Rightarrow a+2b+4=0,a-b+1=0a^2+b^2=5$
$\Rightarrow a=-2,b=-1$