Solution
= Solution
Statement (ii) is \b[false]. The condition is necessary for an idempotent, but it is not sufficient. For example,
$$
\alpha=
\begin{pmatrix}
1&1\\
0&1
\end{pmatrix}
$$
satisfies $(I-\alpha)^2=0$, hence $\alpha(I-\alpha)^2=0$, while
$$
\alpha^2=
\begin{pmatrix}
1&2\\
0&1
\end{pmatrix}\ne\alpha
$$
over, for example, $\mathbb Q$.