Answer
The matrix is orthogonal.
Work Step by Step
Let the matrix be
$A=\left[\begin{array}{cc}
0&1\\
1&0
\end{array}\right]$
Then: $|A|=-1$
We have:
$A^{T}=\left[\begin{array}{cc}
0&1\\
1&0
\end{array}\right]$
and $A^{-1}=(-1) \left[\begin{array}{cc}
0&-1\\
-1&0
\end{array}\right]=\left[\begin{array}{cc}
0&1\\
1&0
\end{array}\right]=A^{T}$
Thus the matrix $A$ is orthogonal.