= Solution
An integer-entry example is
$$
\boxed{A=\begin{pmatrix}0&-1\\1&-1\end{pmatrix}.}
$$
Direct <matrix multiplication> gives $A^2=\bigl(\begin{smallmatrix}-1&1\\-1&0\end{smallmatrix}\bigr)$, so adding $A$ and $I_2$ gives the zero <matrix>. Equivalently, this is the <companion matrix> of $t^2+t+1$ and has the two nonreal cube roots of unity as <eigenvalues>. The direct multiplication is already a complete proof. It also shows why part (e) does not extend to all real square <matrices>: with this $A$, choosing $B=0$ satisfies the original product identity.
Back to article page
