## 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*)

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