Question

The Eigen vectors of the matrix
$\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$ is/are

• A. $(1,0)$
• B. $(0,1)$
• C. $(1,1)$
• D. $(1,-1)$

Answer 1

Answer: (C), (D)

$(1,-1)$

$A=\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$
For eigen values,
|$A-\lambda I$|=0
$\Rightarrow \left|\begin{array}{cc}1-\lambda & -1\\ -1& 1-\lambda \end{array}\right|=0$
$1+{\lambda }^2-2\lambda -1=0$
$\lambda =0,2$
For $\lambda =0$
$AX=0$
$\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$
$\begin{array}{c}x-y=0\\ -x+y=0\end{array}]\Rightarrow y=x$
Let $x=k\Rightarrow y=k$
So, $X=\left[\begin{array}{c}k\\ k\end{array}\right]=k\left[\begin{array}{c}1\\ 1\end{array}\right],kϵR$
Hence option (c) is correct.
By similar logic for
$\lambda =2$, we get $X=k\left[\begin{array}{c}1\\ -1\end{array}\right]$
So option (d) is also correct.