Question :The distinct eigen values of the matrix [1 1 0;1 1 0;0 0 0] are
(1) 0,1
(2) 1, -1
(3) 0,2
(4) 1, 2
Solution :
Let the matrix \[A=\left(\begin{matrix}1&1&0\\1&1&0\\0&0&0\\\end{matrix}\right)\] eigen values of A \[\left|A-\lambda I\right|=0\] Where I is an identity matrix of order 3. \[\left|\left(\begin{matrix}1&1&0\\1&1&0\\0&0&0\\\end{matrix}\right)-\lambda\left(\begin{matrix}1&0&0\\0&1&0\\0&0&1\\\end{matrix}\right)\right|=0\] \[\left|\left(\begin{matrix}1-\lambda&1&0\\1&1-\lambda&0\\0&0&0-\lambda\\\end{matrix}\right)\right|=0\] \[\left(1-\lambda\right)\left(-\lambda\left(1-\lambda\right)-0\right)-1\left(-\lambda-0\right)+0=0 -\lambda\left(1-\lambda\right)^2+\lambda=0\] \[\lambda\left(1+\lambda^2-2\lambda-1\right)=0\] \[\lambda^2\left(\lambda-2\right)=0\] \[\lambda=0\ ,0,\ 2\] Thus distinct eigen values of A are 0, 2. Correct option is (3)