\begin{align*} \left(\hat{I}\otimes{}\left|0\right>\left<1\right|\right) \left(\left|0\right>\otimes{}\left|0\right>+\left|1\right>\otimes{}\left|0\right>\right) & = \left(\left|0\right>\otimes{}\left|0\right>+\left|1\right>\otimes{}\left|0\right>\right)\otimes{} \left[ \left|0\right>\left<1\right| \left(\left|0\right>\otimes{}\left|0\right>+\left|1\right>\otimes{}\left|0\right>\right) \right] \end{align*}