Answer
Since A$A^{T}$ $\ne$ I, the matrix is not orthogonal.
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 = $\begin{bmatrix}
2/3 & 2/3 & 1/3\\
0 & 1/3 & -2/3\\
5/3 & -4/3 & -2/3
\end{bmatrix}$ $A^{T}$ = $\begin{bmatrix}
2/3 & 0 & 5/3\\
2/3 & 1/3 & -4/3\\
1/3 & -2/3 & -2/3
\end{bmatrix}$
A$A^{T}$ = $\begin{bmatrix}
4/9 + 4/9 + 1/9 & 0 + 2/9 - 2/9 & 10/9 - 8/9 - 2/9 \\
0 + 2/9 - 2/9 & 0 + 1/9 + 4/9 & 0 - 4/9 + 4/9 \\
10/9 - 8/9 - 2/9 & 0 - 4/9 + 4/9 & 25/9 +16/9 + 4/9
\end{bmatrix}$ = $\begin{bmatrix}
1 & 0 & 0\\
0 & 5/9 & 0\\
0 & 0 & 5
\end{bmatrix}$
Since A$A^{T}$ $\ne$ I, the matrix is not orthogonal.