Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Digital systems, linear algebra, introduction to formal proof / 5 / d / Solution

Take $A=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$. It has exactly two nonzero entries, both one, but $A^2=I_2\ne A$. Thus it is not an idempotent matrix. Its being an orthogonal matrix does not make it an orthogonal projector.