Math GRE - #8

For $0<t<\pi$, the matrix \[R=\left(\begin{array}{cc}
\cos t & -\sin t\\
\sin t & \cos t\end{array}\right)\] has distinct complex eigenvalues $\lambda_1$ and $\lambda_2$. For what value of $t$ is $\lambda_1+\lambda_2=1$?

• $\frac{\pi}{2}$
• $\frac{\pi}{3}$
• $\frac{\pi}{4}$
• $\frac{\pi}{6}$
• $\frac{2\pi}{3}$

Solution :

Here is a seemingly annoying problem with a simple solution. For any matrix the sum of its eigenvalues is equal to its trace, so we require $\mbox{tr}(R)=2\cos(t)=1$. This is true for $\frac{\pi}{3}$ as $2\cos(\frac{\pi}{3})=2\frac{1}{2}=1.$  Sometimes knowing a superficially useless fact can help avoid a lot of computation.