# 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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## Solution.

### (a) Find the all the eigenvalues of $A$.

Since $A$ is a real matrix and $\frac{-1+\sqrt{3}i}{2}$ is a complex eigenvalue, its conjugate $\frac{-1-\sqrt{3}i}{2}$ is also an eigenvalue of $A$.

As $A$ is a $3\times 3$ matrix, it has one more eigenvalue $\lambda$.

Note that the product of all eigenvalues of $A$ is the determinant of $A$.

Thus, we have

\[\frac{-1+\sqrt{3}i}{2} \cdot \frac{-1-\sqrt{3}i}{2}\cdot \lambda =\det(A)=1.\]
Solving this, we obtain $\lambda=1$.

Therefore, the eigenvalues of $A$ are

\[\frac{-1+\sqrt{3}i}{2}, \frac{-1-\sqrt{3}i}{2}, 1.\]

### (a) Using the Cayley-Hamilton theorem, determine $a, b, c$.

To use the Cayley-Hamilton theorem, we first need to determine the characteristic polynomial $p(t)=\det(A-tI)$ of $A$.

Since we found all the eigenvalues of $A$ in part (a) and the roots of characteristic polynomials are the eigenvalues, we know that

\begin{align*}

p(t)&=-\left(\, t-\frac{-1+\sqrt{3}i}{2} \,\right)\left(\, t-\frac{-1-\sqrt{3}i}{2} \,\right)(t-1) \tag{*}\\

&=-(t^2+t+1)(t-1)\\

&=-t^3+1.

\end{align*}

(Remark that if your definition of the characteristic polynomial is $\det(tI-A)$, then the first negative sign in (*) should be omitted.)

Then the Cayley-Hamilton theorem yields that

\[P(A)=-A^3+I=O,\]
where $O$ is the $3\times 3$ zero matrix.

Hence we have $A^3=I$.

We compute

\begin{align*}

A^{100}=(A^3)^{33}A=I^{33}A=IA=A.

\end{align*}

Thus, we conclude that $a=0, b=1, c=0$.

## Comment.

Observe that we did not use the assumption that $A$ is orthogonal.

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