Answer
The matrix $A$ is not orthogonal.
Work Step by Step
Let the matrix be
$A=\left[\begin{array}{cc}
1&-1\\
-1&-1
\end{array}\right]$
Then: $|A|=-2$
We have:
$A^{T}=\left[\begin{array}{cc}
1&-1\\
-1&-1
\end{array}\right]$
and $A^{-1}= (-1/2)* \left[\begin{array}{cc}
-1&1\\
1&1
\end{array}\right]=(1/2)* \left[\begin{array}{cc}
1&-1\\
-1&-1
\end{array}\right] =(1/2)* A^{T} \neq A^{T}$
Thus the matrix $A$ is not orthogonal.
