Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2004 / Paper 2 / 3 / b / iii / Solution

Write the general real symmetric matrix as $E_2=\left(\begin{array}{cc}r& t\\ t& u\end{array}\right)$. With the specified $E_1$, anticommutation readsHence $t=0$ and $u=-r$. The remaining condition $E_2^2=I$ gives $r^2=1$, leaving exactlyBoth matrices directly satisfy all the conditions.