# 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)

Sponsored Links

## 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.

Sponsored Links

### More from my site

• Find the Eigenvalues and Eigenvectors of the Matrix $A^4-3A^3+3A^2-2A+8E$. Let $A=\begin{bmatrix} 1 & -1\\ 2& 3 \end{bmatrix}.$ Find the eigenvalues and the eigenvectors of the matrix $B=A^4-3A^3+3A^2-2A+8E.$ (Nagoya University Linear Algebra Exam Problem)   Hint. Apply the Cayley-Hamilton theorem. That is if $p_A(t)$ is the […]
• A Square Root Matrix of a Symmetric Matrix Answer the following two questions with justification. (a) Does there exist a $2 \times 2$ matrix $A$ with $A^3=O$ but $A^2 \neq O$? Here $O$ denotes the $2 \times 2$ zero matrix. (b) Does there exist a $3 \times 3$ real matrix $B$ such that $B^2=A$ […]
• Find All the Eigenvalues of 4 by 4 Matrix Find all the eigenvalues of the matrix $A=\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 &0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{bmatrix}.$ (The Ohio State University, Linear Algebra Final Exam Problem)   Solution. We compute the […]
• Diagonalize a 2 by 2 Matrix $A$ and Calculate the Power $A^{100}$ Let $A=\begin{bmatrix} 1 & 2\\ 4& 3 \end{bmatrix}.$ (a) Find eigenvalues of the matrix $A$. (b) Find eigenvectors for each eigenvalue of $A$. (c) Diagonalize the matrix $A$. That is, find an invertible matrix $S$ and a diagonal matrix $D$ such that […]
• How to Calculate and Simplify a Matrix Polynomial Let $T=\begin{bmatrix} 1 & 0 & 2 \\ 0 &1 &1 \\ 0 & 0 & 2 \end{bmatrix}$. Calculate and simplify the expression $-T^3+4T^2+5T-2I,$ where $I$ is the $3\times 3$ identity matrix. (The Ohio State University Linear Algebra Exam) Hint. Use the […]
• Find Eigenvalues, Eigenvectors, and Diagonalize the 2 by 2 Matrix Consider the matrix $A=\begin{bmatrix} a & -b\\ b& a \end{bmatrix}$, where $a$ and $b$ are real numbers and $b\neq 0$. (a) Find all eigenvalues of $A$. (b) For each eigenvalue of $A$, determine the eigenspace $E_{\lambda}$. (c) Diagonalize the matrix $A$ by finding a […]
• A Diagonalizable Matrix which is Not Diagonalized by a Real Nonsingular Matrix Prove that the matrix $A=\begin{bmatrix} 0 & 1\\ -1& 0 \end{bmatrix}$ is diagonalizable. Prove, however, that $A$ cannot be diagonalized by a real nonsingular matrix. That is, there is no real nonsingular matrix $S$ such that $S^{-1}AS$ is a diagonal […]
• Rotation Matrix in Space and its Determinant and Eigenvalues For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by $A=\begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta &\cos\theta &0 \\ 0 & 0 & 1 \end{bmatrix}.$ (a) Find the determinant of the matrix $A$. (b) Show that $A$ is an […]

#### You may also like...

This site uses Akismet to reduce spam. Learn how your comment data is processed.

##### If $A$ is a Skew-Symmetric Matrix, then $I+A$ is Nonsingular and $(I-A)(I+A)^{-1}$ is Orthogonal

Let $A$ be an $n\times n$ real skew-symmetric matrix. (a) Prove that the matrices $I-A$ and $I+A$ are nonsingular. (b)...

Close