Question

The Eigen value of the matrix $A=\left[\begin{array}{ccc}2& 1& 1\\ 2& 3& 4\\ -1& -1& -2\end{array}\right]$ are

• A. $2,1,1$
• B. $2,3,-2$
• C. $-1,1,3$
• D. $noneofthese$

Answer 1

The correct option is C $-1,1,3$

$A=\left[\begin{array}{ccc}2& 1& 1\\ 2& 3& 4\\ -1& -1& -2\end{array}\right]$

$|A-\lambda I|=0$

$\left[\begin{array}{ccc}2-\lambda & 1& 1\\ 2& 3-\lambda & 4\\ -1& -1& -2-\lambda \end{array}\right]=0$

$\left(2-\lambda \right)\left[\left(3-\lambda \right)\left(-2-\lambda \right)+4\right]-1\left[-4-2\lambda +4\right]+\left[-2+3-\lambda \right]=0$

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

$-{\lambda }^3+3{\lambda }^2+\lambda -3=0$

${\lambda }^3-3{\lambda }^2-\lambda +3=0$

$\left(\lambda -1\right)\left({\lambda }^2-2\lambda -3\right)=0$

$\left(\lambda -1\right)\left(\lambda +1\right)\left(\lambda -3\right)=0$

$\lambda =-1,1,3$

Therefore, Eigen values of matrix $A$ are $-1,1,3$