# True or False: Eigenvalues of a Real Matrix Are Real Numbers

## Problem 67

Answer the following questions regarding eigenvalues of a real matrix.

(a) True or False. If each entry of an $n \times n$ matrix $A$ is a real number, then the eigenvalues of $A$ are all real numbers.
(b) Find the eigenvalues of the matrix
$B=\begin{bmatrix} -2 & -1\\ 5& 2 \end{bmatrix}.$

(The Ohio State University, Linear Algebra Exam)

## Hint.

Consider a $2\times 2$ matrix.
Then the eigenvalues are solutions of a quadratic polynomial.

Does a quadratic polynomial always have real solutions?

## Solution.

### (a) True or False. If each entry of an $n \times n$ matrix $A$ is a real number, then the eigenvalues of $A$ are all real numbers.

False. In general, a real matrix can have a complex number eigenvalue. In fact, the part (b) gives an example of such a matrix.

### (b) Find the eigenvalues of the matrix

The characteristic polynomial for $B$ is
$\det(B-tI)=\begin{bmatrix} -2-t & -1\\ 5& 2-t \end{bmatrix}=t^2+1.$

The eigenvalues are the solutions of the characteristic polynomial. Thus solving $t^2+1=0$, we obtain eigenvalues $\pm i$, where $i=\sqrt{-1}$.
Thus the eigenvalue of a real matrix $B$ is pure imaginary numbers $\pm i$.

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