Exercise 10.A.16

Proof. We give a counterexample. Define $T\in L({ℝ}^2)$ by

The matrix of $T$ with respect to standard basis of ${ℝ}^2$ is

Hence $\mathrm{trace}<!--\; FUNCTION\; APPLICATION\; -->T=0$. However, $T^2=-I$, which implies that $\mathrm{trace}<!--\; FUNCTION\; APPLICATION\; -->T^2=-2$. □