If  $A = \left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ a& 2& b\end{array}\right]$ is a matrix satisfying $AA\text{'} = 9I_3,$ then  the value of $\left|a\right| + \left|b\right|.$

Since $$\begin{gathered} AA\text{'} = 9I_3 \\ \Rightarrow \left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ a& 2& b\end{array}\right] \left[\begin{array}{ccc}1& 2& a\\ 2& 1& 2\\ 2& -2& b\end{array}\right] = 9\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}9& 0& a+2b+4\\ 0& 9& 2a-2b+2\\ a+2b+4& 2a-2b+2& a^2+b^2 +4\end{array}\right] = \left[\begin{array}{ccc}9& 0& 0\\ 0& 9& 0\\ 0& 0& 9\end{array}\right] \end{gathered}$$

Equating the corresponding elements, we get 

$$\begin{gathered} a + 2b +4 = 0.....\left(i\right) \\ 2a - 2b +2 = 0 ....\left(ii\right) \end{gathered}$$

and $a^2 + b^2 +4 = 9 .....\left(iii\right)$

From Eqs. $\left(i\right)$and $\left(ii\right)$, we get 

$a = -2$ and $b = -1$

Hence, $\left|a\right| + \left|b\right| = \left|-2\right| + \left|-1\right| = 2 +1 = 3$