Question

The matrix $\left[\begin{array}{cc}2& -4\\ 4& -2\end{array}\right]$ has

• A. Real eigen values and eigen vectors
• B. Real eigen values but complex eigen vectors
• C. Complex eigen values but real eigen vectors
• D. Complex eigen values and eigen vectors

Answer 1

The correct option is D Complex eigen values and eigen vectors

A = $\left[\begin{array}{cc}2& -4\\ 4& -2\end{array}\right]$ $\begin{array}{c}{\lambda }_1+{\lambda }_2=Trace\left(A\right)=0\\ {\lambda }_1.{\lambda }_2=|A|=12\end{array}$

$\Rightarrow$ ${\lambda }_1$ = -${\lambda }_2$ and (${\lambda }_1$)($-{\lambda }_1$) = 12

${\lambda }_1^2$ = -12 $\Rightarrow$ ${\lambda }_1$ = $2\sqrt{3}i$

and ${\lambda }_2$ = -$2\sqrt{3}i$

Hence, Eigen values are complex and consequently Eigen vectors will also be complex.