Question

The product of the non-zero eigen values of the matrix $\left[\begin{array}{ccccc}1& 0& 0& 0& 1\\ 0& 1& 1& 1& 0\\ 0& 1& 1& 1& 0\\ 0& 1& 1& 1& 0\\ 1& 0& 0& 0& 1\end{array}\right]$ is_____.

• A. 6

Answer 1

The correct option is A 6
Let the given matrix be 'A'
Characteristic Equation is $|A-\lambda I|=0$
$\Rightarrow \left|\begin{array}{ccccc}1-\lambda & 0& 0& 0& 1\\ 0& 1-\lambda & 1& 1& 0\\ 0& 1& 1-\lambda & 1& 0\\ 0& 1& 1& 1-\lambda & 0\\ 1& 0& 0& 0& 1-\lambda \end{array}\right|=0$

Expanding by $1^{st}$ row
$(1-\lambda )\left|\begin{array}{cccc}1-\lambda & 1& 1& 0\\ 1& 1-\lambda & 1& 0\\ 1& 1& 1-\lambda & 0\\ 0& 0& 0& 1-\lambda \end{array}\right|+\left|\begin{array}{cccc}0& 1-\lambda & 1& 1\\ 0& 1& 1-\lambda & 1\\ 0& 1& 1& 1-\lambda \\ 1& 0& 0& 0\end{array}\right|=0$
or $(1-\lambda {)}^2\left|\begin{array}{ccc}1-\lambda & 1& 1\\ 1& 1-\lambda & 1\\ 1& 1& 1-\lambda \end{array}\right|-(1)\left|\begin{array}{ccc}1-\lambda & 1& 1\\ 1& 1-\lambda & 1\\ 1& 1& 1-\lambda \end{array}\right|=0$ ...(1)

Let $B=\left[\begin{array}{ccc}1-\lambda & 1& 1\\ 1& 1-\lambda & 1\\ 1& 1& 1-\lambda \end{array}\right]$

$\Rightarrow$ $|B|=-{\lambda }^2(\lambda -3)$

So $(1-\lambda {)}^2|B|-(1)|B|=0$ [using (1)]

$|B|\left[\right(1-\lambda {)}^2-1]=0$

$-{\lambda }^2(\lambda -3)\left[\lambda \right(\lambda -2\left)\right]=0$
$\Rightarrow {\lambda }^3(\lambda -2)(\lambda -3)=0$
So $\lambda =0,0,0,2,3$
Hence required product $=2\times 3=6$