Answer
The matrix $A$ is orthogonal.
Work Step by Step
Since $|A|=-1$, then $A$ is invertible.
$A=\left [\begin{array}{ccc}
1 & 0 & 0 \\
0 &0&1\\
0&1&0
\end {array} \right] $
The augmented matrix of $A$ is $A=\left [\begin{array}{ccc}
-1 & 0 & 0 \\
0 &0&-1\\
0&-1&0
\end {array} \right] $
and $A^{-1}=(1/|A|)*adjA=(-1)* \left [\begin{array}{ccc}
-1 & 0 & 0 \\
0 &0&-1\\
0&-1&0
\end {array} \right] =\left [\begin{array}{ccc}
1 & 0 & 0 \\
0 &0&1\\
0&1&0
\end {array} \right] $
we have from matrix $A$
$A^{T}=\left [\begin{array}{ccc}
1 & 0 & 0 \\
0 &0&1\\
0&1&0
\end {array} \right]$
Therefore, $ A^{T}= A^{-1}$
Thus the matrix $A$ is orthogonal.
