Question

The eigen values of the following matrix are
$\left[\begin{array}{ccc}-1& 3& 5\\ -3& -1& 6\\ 0& 0& 3\end{array}\right]$

• A. $3,3+5j,6-j$
• B. $-6+5j,3+j,3-j$
• C. $3+j,3-j,5+j$
• D. $3,-1+3j,-1-3j$

Answer 1

The correct option is D $3,-1+3j,-1-3j$

Characteristic equation is
$|A-\lambda I|=0$
$\Rightarrow \left|\begin{array}{ccc}-1-\lambda & 3& 5\\ -3& -1-\lambda & 6\\ 0& 0& 3-\lambda \end{array}\right|=0$
${\lambda }^3-{\lambda }^2+4\lambda -30=0$
$(\lambda -3)({\lambda }^2+2\lambda +10)=0$
$\lambda =3,-1\pm 3j$
Hence eigen values are $3,-1+3j,-1-3j$