[amp_mcq option1=”real eigenvalues and eigenvectors” option2=”real eigenvalues but complex eigenvectors” option3=”complex eigenvalues but real eigenvectors” option4=”complex eigenvalues and eigenvectors” correct=”option3″]
The correct answer is $\boxed{\text{(B) real eigenvalues but complex eigenvectors}}$.
To find the eigenvalues and eigenvectors of a matrix, we can use the following formula:
$$\lambda v = A v$$
where $\lambda$ is the eigenvalue, $v$ is the eigenvector, and $A$ is the matrix.
In this case, we have the following matrix:
$$A = \left( {\begin{array}{*{20}{c}} 2&{ – 4} \ 4&{ – 2} \end{array}} \right)$$
Substituting this into the formula, we get the following equations:
$$\begin{align}
2 v_1 – 4 v_2 &= \lambda v_1 \
4 v_1 – 2 v_2 &= \lambda v_2
\end{align}$$
Solving these equations, we get the following eigenvalues:
$$\lambda = 2 \pm 2 i$$
and the following eigenvectors:
$$v_1 = \left( {\begin{array}{c} 1 \ -i \end{array}} \right) \text{ and } v_2 = \left( {\begin{array}{c} 1 \ i \end{array}} \right)$$
As you can see, the eigenvalues are real, but the eigenvectors are complex. This is because the matrix $A$ is not a real matrix. It is a complex matrix.
Therefore, the correct answer is $\boxed{\text{(B) real eigenvalues but complex eigenvectors}}$.