Eigen value of the following matrix

Question

The eigen value of the following matrix is $\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}$ 

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

Answer 1

$let  A = \begin{bmatrix} 1 &1 &1 \\ 1 &1 &1 \\ 1 &1 &1 \end{bmatrix} $

Characterstatics equation of this matrix is |A-$\lambda $I|=0

=> $\begin{vmatrix} 1-\lambda &1 &1 \\ 1 &1-\lambda &1 \\ 1 &1 &1-\lambda \end{vmatrix}$ = 0

=>$(1-\lambda)\begin{vmatrix} 1-\lambda &1 \\ 1 &1-\lambda \end{vmatrix}$  - $(1)\begin{vmatrix} 1 &1 \\ 1\ &1-\lambda\end{vmatrix}$ + (1)$\begin{vmatrix} 1 &1-\lambda \\ 1 &1 \end{vmatrix}$ = 0

=>(1-$\lambda$ ) { (1-$\lambda$)$^{2}$ - 1 } - $(1 -\lambda - 1 ) + (1 - 1 + \lambda ) = 0$

=>$(1-\lambda)(1 + \lambda ^{2} - 2\lambda -1) +\lambda +\lambda =0$

=>$(1-\lambda)(\lambda ^{2} - 2\lambda ) + 2 \lambda = 0$

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

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

=>$\lambda ^{3}  - 3\lambda ^{2} = 0$

=>$\lambda^{2}(\lambda - 1) =0$

=>So,$\lambda$ $^{2}$ = 0  and  ($\lambda$ - 3) =0

answer is => $\lambda$ = 0,0 and 3

option (C) is the correct answer

Comments

You can also solve using Eigenvalue property:

(1)Sum of all Eigenvalue = sum of the leading diagonal element

(2)  |A| = product of all Eigenvalue

I can take A = $\begin{bmatrix} 1 &1 &1 \\ 1 &1 &1 \\ 1 &1 &1 \end{bmatrix}$

=> |A| = $\begin{vmatrix} 1 &1 &1 \\ 1 &1 &1 \\ 1 &1 &1 \end{vmatrix}$ = 0

 

property(1):sum of leading diagonal element = 1 + 1 + 1 =3 = sum of all Eigenvalue

Property(2):|A| = 0 = Product of all Eigenvalue

You can easily see option (C) is the correct.

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Yeah I got it thnxx laksman