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)

 
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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.

  1. Find All the Eigenvalues of 4 by 4 Matrix (This page)
  2. Find a Basis of the Eigenspace Corresponding to a Given Eigenvalue
  3. Diagonalize a 2 by 2 Matrix if Diagonalizable
  4. Find an Orthonormal Basis of the Range of a Linear Transformation
  5. The Product of Two Nonsingular Matrices is Nonsingular
  6. Determine Whether Given Subsets in ℝ4 R 4 are Subspaces or Not
  7. Find a Basis of the Vector Space of Polynomials of Degree 2 or Less Among Given Polynomials
  8. Find Values of $a , b , c$ such that the Given Matrix is Diagonalizable
  9. Idempotent Matrix and its Eigenvalues
  10. Diagonalize the 3 by 3 Matrix Whose Entries are All One
  11. Given the Characteristic Polynomial, Find the Rank of the Matrix
  12. Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$
  13. Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$

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