Question

A system matrix is given as follows :
$A=\left[\begin{array}{ccc}0& 1& -1\\ -6& -11& 6\\ -6& -11& 5\end{array}\right]$
The absolute value of the ratio of the maximum eigen value to the minimum eigen value is __

• A. 0.333

Answer 1

The correct option is A 0.333
Characteristic equation is
$|A-\lambda I|=0$
$\Rightarrow \left|\begin{array}{ccc}-\lambda & 1& -1\\ -6& -11-\lambda & 6\\ -6& -11& 5-\lambda \end{array}\right|=0$
$\Rightarrow {\lambda }^3+6{\lambda }^2+11\lambda +6=0$
$\Rightarrow \lambda =-1,-2,-3$
${\lambda }_{max=-1}$
${\lambda }_{min=-3}$
$\left|\begin{array}{c}\frac{{\lambda }_{max}}{{\lambda }_{min}}\end{array}\right|=|\frac{-1}{-3}|=\frac{1}{3}$