# Find All the Eigenvalues of 4 by 4 Matrix ## 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

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###### More in Linear Algebra ##### The Determinant of a Skew-Symmetric Matrix is Zero

Prove that the determinant of an $n\times n$ skew-symmetric matrix is zero if $n$ is odd.

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