Question

The eigen values of $\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$ are

• A. $0,0,0$
• B. $0,0,1$
• C. $0,0,3$
• D. $1,1,1$

Answer 1

The correct option is C $0,0,3$

Characteristic equation of matrix is $|A-\lambda I|$=0
$\left|\begin{array}{ccc}1-\lambda & 1& 1\\ 1& 1-\lambda & 1\\ 1& 1& 1-\lambda \end{array}\right|$=0
$(1-\lambda )\left[\right(1-\lambda {)}^2-1\left)\right]-\left[\right(1-\lambda )-1]+1[1-(1-\lambda \left)\right]=0$
$(1-\lambda {)}^3-(1-\lambda )+2\lambda =0$
${\lambda }^3-3{\lambda }^2=0$
${\lambda }^2(\lambda -3)=0$
$\lambda =0,0,3$