Science:Math Exam Resources/Courses/MATH152/April 2015/Question B 6 (a)/Solution 2

Following the second hint, Since matrix $A$ is a orthogonal rotation matrix, therefore it has determinant $\pm 1$. For this question, the determinant is $1$.

Out of three eigenvalues, one of them must be 1 from the property of an orthogonal rotation matrix. Now suppose the left two eigenvalues are $z_1$ and $z_2$. Since the sum of the eigenvalues is the trace of the matrix, and since their product is the determinant, we have $1+z_1+z_2=1$ and $z_1\cdot z_2=1$.

$z_1=i,z_2=-i$

answer: $1,i,-i$