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

Ohio State University exam problems and solutions in mathematics

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)

LoadingAdd to solve later

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 $\pm i$.


LoadingAdd to solve later

Sponsored Links

More from my site

You may also like...

Leave a Reply

Your email address will not be published. Required fields are marked *

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

More in Linear Algebra
Ohio State University exam problems and solutions in mathematics
Linear Independent Vectors, Invertible Matrix, and Expression of a Vector as a Linear Combinations

Consider the matrix \[A=\begin{bmatrix} 1 & 2 & 1 \\ 2 &5 &4 \\ 1 & 1 & 0 \end{bmatrix}.\]...

Close