Question

A system matrix is given as follows
$A=\left|\begin{array}{ccc}0& 2& -2\\ -12& -22& 12\\ -12& -22& 10\end{array}\right|$

The value of the ratio of the absolute maximum eigen value to
the absolute minimum eigen value is______.

• A. 3

Answer 1

The correct option is A 3
Characteristic equation is $|A-\lambda I|=0$

$\left|\begin{array}{ccc}-\lambda & 2& -2\\ -12& -22-\lambda & 12\\ -12& -22& 10-\lambda \end{array}\right|-0$

$\Rightarrow -\lambda [-220+22\lambda -10\lambda +{\lambda }^2+264]-2[-120+12\lambda +144]-2[264-264-12\lambda ]=0$

$\Rightarrow -\lambda ({\lambda }^2+12\lambda +44)-2(12\lambda +24)+24\lambda =0$

$\Rightarrow -{\lambda }^3-12{\lambda }^2-44\lambda -24\lambda -48+24\lambda =0$

$\Rightarrow -{\lambda }^3-12{\lambda }^2-44\lambda -48=0$

$\Rightarrow {\lambda }^3+12{\lambda }^2+44\lambda +48=0$

$\Rightarrow (\lambda +2)(\lambda +4)(\lambda +6)=0$

$\Rightarrow \lambda =-2,-4-6$

Ratio $=\frac{6}{2}=3$