Answer
Since A$A^{T}$ = I, the matrix is orthogonal.
$A^{-1}$= $\begin{bmatrix}
1/3 & 2/3 & 2/3\\
2/3 & 1/3 & -2/3\\
2/3 & -2/3 & 1/3
\end{bmatrix}$
Work Step by Step
If a matrix is orthogonal, then A$A^{T}$ = I. If a matrix is orthogonal, then the inverse of the matrix is equal to $A^{T}$.
First, we determine if A$A^{T}$ = I.
A = $\begin{bmatrix}
1/3 & 2/3 & 2/3\\
2/3 & 1/3 & -2/3\\
2/3 & -2/3 & 1/3
\end{bmatrix}$ $A^{T}$ = $\begin{bmatrix}
1/3 & 2/3 & 2/3\\
2/3 & 1/3 & -2/3\\
2/3 & -2/3 & 1/3
\end{bmatrix}$
A$A^{T}$ = $\begin{bmatrix}
1/9 + 4/9 + 4/9 & 2/9 + 2/9 - 4/9 & 2/9 - 4/9 + 2/9 \\
2/9 + 2/9 - 4/9 & 4/9 + 1/9 + 4/9 & 4/9 - 2/9 - 2/9 \\
2/9 -4/9 2/9 & 4/9 - 2/9 - 2/9 & 4/9 +1/9 + 1/9
\end{bmatrix}$ = $\begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{bmatrix}$
Since A$A^{T}$ = I, the matrix is orthogonal.
Since the matrix is orthogonal. $A^{-1}$ = $A^{T}$, so
$A^{-1}$= $\begin{bmatrix}
1/3 & 2/3 & 2/3\\
2/3 & 1/3 & -2/3\\
2/3 & -2/3 & 1/3
\end{bmatrix}$
