Let $A$ be a $2\times 2$ matrix such that $A^2+I=0$. If $A$ has real entries, find the trace of $A$.

Solution

1. The characteristic equation of $A$ is ${\lambda }^2+1=0$.
2. The eigenvalues are $\lambda =i$ and $\lambda =-i$.
3. The trace of a matrix is the sum of its eigenvalues.
4. $$\begin{array}{rl}\text{tr}(A)& =i+(-i)\\ & =0\end{array}$$. Hence the answer is (A).