Mathematics - Matrices Question with Solution | TestHub

Given,$A{A}^T=9I$

$\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ a& 2& b\end{array}\right]\left[\begin{array}{ccc}1& 2& a\\ 2& 1& 2\\ 2& -2& b\end{array}\right]=9I$

$\Rightarrow \left[\begin{array}{ccc}9& 0& a+4+2b\\ 0& 9& 2a+2-2b\\ a+4+2b& 2a+2-2b& a^2+4+b^2\end{array}\right]=\left[\begin{array}{ccc}9& 0& 0\\ 0& 9& 0\\ 0& 0& 9\end{array}\right]$

Equation$a+4+2b=0$$\Rightarrow a+2b=-4 ...\left(1\right)$

$2a+2-2b=0$$\Rightarrow 2a-2b=-2 ...\left(2\right)$

$\&a^2+4+b^2=9\Rightarrow a^2+b^2=5 ...\left(3\right)$

On solving, we get$a=-2,b=-1$