Answer
No.
Work Step by Step
$\begin{bmatrix}
0& 1 \\
1&0\\
\end{bmatrix} \begin{bmatrix}
0& 1 \\
1&0\\
\end{bmatrix}= \begin{bmatrix}
0\cdot0+1\cdot1& ? \\
?&?\\
\end{bmatrix}=\begin{bmatrix}
1& ? \\
?&?\\
\end{bmatrix}$
We see that $A^2\ne A$.
Thus we can see that the matrix is definitely not idempotent.
