Use the Cayley-Hamilton Theorem to Compute the Power $A^{100}$
Problem 471
Let $A$ be a $3\times 3$ real orthogonal matrix with $\det(A)=1$.
(a) If $\frac{-1+\sqrt{3}i}{2}$ is one of the eigenvalues of $A$, then find the all the eigenvalues of $A$.
(b) Let
\[A^{100}=aA^2+bA+cI,\]
where $I$ is the $3\times 3$ identity matrix.
Using the Cayley-Hamilton theorem, determine $a, b, c$.
(Kyushu University, Linear Algebra Exam Problem)
Read solution