Answer
The matrix $A$ is not orthogonal.
Work Step by Step
Let the matrix be
$A=\left[\begin{array}{cc}
1&0\\
1&1
\end{array}\right]$
Then: $|A|=1$
We have:
$A^{T}=\left[\begin{array}{cc}
1&1\\
0&1
\end{array}\right]$
and $A^{-1}= \left[\begin{array}{cc}
1&0\\
-1&1
\end{array}\right]\neq A^{T}$
Thus the matrix $A$ is not orthogonal.