Linear Algebra

Find All the Eigenvalues of 4 by 4 Matrix

Ohio State University exam problems and solutions in mathematics

Problem 475

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)

Add to solve later

Solution.

We compute the characteristic polynomial $p(t)$ of the matrix $A$ as follows.
We have
\begin{align*}
p(t)&=\det(A-tI)\\
&=\begin{vmatrix}
-t & 1 & 0 & 0 \\
0 &-t & 1 & 0 \\
0 & 0 & -t & 1 \\
1 & 0 & 0 & -t
\end{vmatrix}\\[6pt] &=-t\begin{vmatrix}
-t & 1 & 0 \\
0 &-t &1 \\
0 & 0 & -t
\end{vmatrix}
-\begin{vmatrix}
0 & 1 & 0 \\
0 &-t &1 \\
1 & 0 & -t
\end{vmatrix} \tag{*}
\end{align*}
by the first row cofactor expansion.
The left determinant of the $3\times 3$ matrix in (*) is $(-t)^3$ since it is a diagonal matrix.

We apply the first column cofactor expansion to the right determinant in (*) and obtain
\begin{align*}
\begin{vmatrix}
0 & 1 & 0 \\
0 &-t &1 \\
1 & 0 & -t
\end{vmatrix} =\begin{vmatrix}
1 & 0\\
-t& 1
\end{vmatrix}=1.
\end{align*}

It follows from (*) that
\begin{align*}
p(t)&=(-t)(-t)^3-1=t^4-1.
\end{align*}

The eigenvalues of $A$ are the roots of the characteristic polynomial $p(t)$.
Solving $t^4-1=0$, we obtain the eigenvalues
\[\pm 1, \pm i,\] where $i=\sqrt{-1}$.
Note that $t^4-1=(t-1)(t+1)(t-i)(t+i)$.

Final Exam Problems and Solution. (Linear Algebra Math 2568 at the Ohio State University)

This problem is one of the final exam problems of Linear Algebra course at the Ohio State University (Math 2568).

The other problems can be found from the links below.

Add to solve later

More from my site

Given All Eigenvalues and Eigenspaces, Compute a Matrix Product Let $C$ be a $4 \times 4$ matrix with all eigenvalues $\lambda=2, -1$ and eigensapces \[E_2=\Span\left \{\quad \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} \quad\right \} \text{ and } E_{-1}=\Span\left \{ \quad\begin{bmatrix} 1 \\ 2 \\ 1 \\ 1 […]
$Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$$ Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$ Let \[A=\begin{bmatrix} 1 & -14 & 4 \\ -1 &6 &-2 \\ -2 & 24 & -7 \end{bmatrix} \quad \text{ and }\quad \mathbf{v}=\begin{bmatrix} 4 \\ -1 \\ -7 \end{bmatrix}.\] Find $A^{10}\mathbf{v}$. You may use the following information without proving […]
Diagonalize a 2 by 2 Matrix if Diagonalizable Determine whether the matrix \[A=\begin{bmatrix} 1 & 4\\ 2 & 3 \end{bmatrix}\] is diagonalizable. If so, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$. (The Ohio State University, Linear Algebra Final Exam […]
Find the Nullity of the Matrix $A+I$ if Eigenvalues are $1, 2, 3, 4, 5$ Let $A$ be an $n\times n$ matrix. Its only eigenvalues are $1, 2, 3, 4, 5$, possibly with multiplicities. What is the nullity of the matrix $A+I_n$, where $I_n$ is the $n\times n$ identity matrix? (The Ohio State University, Linear Algebra Final Exam […]
Determine Dimensions of Eigenspaces From Characteristic Polynomial of Diagonalizable Matrix Let $A$ be an $n\times n$ matrix with the characteristic polynomial \[p(t)=t^3(t-1)^2(t-2)^5(t+2)^4.\] Assume that the matrix $A$ is diagonalizable. (a) Find the size of the matrix $A$. (b) Find the dimension of the eigenspace $E_2$ corresponding to the eigenvalue […]
Find a Basis of the Eigenspace Corresponding to a Given Eigenvalue Let \[A=\begin{bmatrix} 1 & 2 & 1 \\ -1 &4 &1 \\ 2 & -4 & 0 \end{bmatrix}.\] The matrix $A$ has an eigenvalue $2$. Find a basis of the eigenspace $E_2$ corresponding to the eigenvalue $2$. (The Ohio State University, Linear Algebra Final Exam […]
Diagonalize the 3 by 3 Matrix Whose Entries are All One Diagonalize the matrix \[A=\begin{bmatrix} 1 & 1 & 1 \\ 1 &1 &1 \\ 1 & 1 & 1 \end{bmatrix}.\] Namely, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$. (The Ohio State University, Linear Algebra Final Exam […]
Find Values of $a, b, c$ such that the Given Matrix is Diagonalizable For which values of constants $a, b$ and $c$ is the matrix \[A=\begin{bmatrix} 7 & a & b \\ 0 &2 &c \\ 0 & 0 & 3 \end{bmatrix}\] diagonalizable? (The Ohio State University, Linear Algebra Final Exam Problem) Solution. Note that the […]