Solution
= Solution
Statement (iv) is \b[false] because $\alpha\beta=0$ need not imply $\beta\alpha=0$. For example, over any field take
$$
\alpha=
\begin{pmatrix}
1&0\\
0&0
\end{pmatrix},
\qquad
\beta=
\begin{pmatrix}
0&0\\
1&1
\end{pmatrix}.
$$
Both maps are idempotent and $\alpha\beta=0$, but
$$
\beta\alpha=
\begin{pmatrix}
0&0\\
1&0
\end{pmatrix}\ne0.
$$
Consequently
$$
\boxed{(\alpha+\beta)^2=\alpha+\beta+\beta\alpha\ne\alpha+\beta.}
$$