$A=\begin{bmatrix}1 &2 &2\\2&1&-2\\a&2&b\end{bmatrix}$ is a matrix satisfying the
equation $A A^T=9I$, where I is 3 × 3 matrix, then the ordered pair (a, b) is equal to

(-2, -1)

We have,

$A A^T=9I$

$⇒\begin{bmatrix}1 &2 &2\\2&1&-2\\a&2&b\end{bmatrix}\begin{bmatrix}1 &2 &a\\2&1&2\\2&-2&b\end{bmatrix}=\begin{bmatrix}9 &0 &0\\0&9&0\\0&0&9\end{bmatrix}$

$⇒\begin{bmatrix}9 &0 &a+4+2b\\0&9&2a+2-2b\\a+4+2b&2a+2-2b&a^2+4+b^2\end{bmatrix}=\begin{bmatrix}9 &0 &0\\0&9&0\\0&0&9\end{bmatrix}$

$⇒a+4+2b=0,2a+2-2b = 0$ and $a^2+4+b^2=9$

$⇒a=-2,b=-1$

Hence, $(a,b)=(-2,-1)$.