Question

In the given matrix $\left[\begin{array}{ccc}1& -1& 2\\ 0& 1& 0\\ 1& 2& 1\end{array}\right]$, one of the eigen value is 1. The eigen vectors corresponding to the eigen value 1 are

• A. $\alpha (4,2,1)|\alpha \ne 0,\alpha ϵR$
• B. $\alpha (-4,2,1)|\alpha \ne 0,\alpha ϵR$
• C. $\alpha (\sqrt{2},0,1)|\alpha \ne 0,\alpha ϵR$
• D. $\alpha (-\sqrt{2},2,1)|\alpha \ne 0,\alpha ϵR$

Answer 1

The correct option is B $\alpha (-4,2,1)|\alpha \ne 0,\alpha ϵR$

Let A = $\left[\begin{array}{ccc}1& -1& -2\\ 0& 1& 0\\ 1& 2& 1\end{array}\right]$

Consider AX = $\lambda$X

$\Rightarrow$ (A - $\lambda I$)X = O

For $\lambda$ = 1, $(A-I)X$ = O

So $\left[\begin{array}{ccc}0& -1& 2\\ 0& 0& 0\\ 1& 2& 0\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

$\left[\begin{array}{ccc}1& 2& 0\\ 0& -1& 2\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

$\Rightarrow$ x + 2y = 0 and -y + 2z = 0

Let z = k. Then y = 2k and x = -4k

So X = $\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}-4k\\ 2k\\ k\end{array}\right]=k\left[\begin{array}{c}-4\\ 2\\ 1\end{array}\right]$