Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Digital systems, linear algebra, introduction to formal proof resit / 5 / f / Solution

An integer-entry example isDirect matrix multiplication gives $A^2=\left(\begin{array}{cc}-1& 1\\ -1& 0\end{array}\right)$, 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.