If $\mathrm{A}=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ \mathrm{a}& 2& \mathrm{b}\end{array}\right]$is a matrix satisfying the equation AA^T = 9I, where I is 3 × 3 identity matrix, then a+b is

$\begin{array}{l}{\mathrm{AA}}^{\mathrm{T}}=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ \mathrm{a}& 2& \mathrm{b}\end{array}\right]\left[\begin{array}{ccc}1& 2& \mathrm{a}\\ 2& 1& 2\\ 2& -2& \mathrm{b}\end{array}\right]\\ \Rightarrow \left[\begin{array}{ccc}9& 0& a+2b+4\\ 0& 9& 2a+2-2b\\ a+2b+4& 2a+2-2b& a^2+4+b^2\end{array}\right]\\ \text{ Now using given condition, }{\mathrm{AA}}^{\mathrm{T}}=9\mathrm{I}\\ \Rightarrow \left[\begin{array}{ccc}9& 0& a+2b+4\\ 0& 9& 2a+2-2b\\ a+2b+4& 2a+2-2b& a^2+4+b^2\end{array}\right]=9\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\\ a+2b+4=0 and\\ {\mathrm{a}}^2+4+{\mathrm{b}}^2=9\Rightarrow \mathrm{a}=-2,\mathrm{b}=-1\end{array}$