Question

Identify which one of the following is an eigen vector of the matrix $A=\left|\begin{array}{cc}1& 0\\ -1& -2\end{array}\right|$

• A. $[-11{]}^T$
• B. $[3-1{]}^T$
• C. $[1-1{]}^T$
• D. $[-21{]}^T$

Answer 1

The correct option is B $[3-1{]}^T$

Eigen values are $1,2,(\because$ it is $L.T.M.)$
For $\lambda =1$, eigen vectors are given by $(A-\lambda I)x=0$
or $(A-I)X=0$
or $\left[\begin{array}{cc}0& 0\\ -1& -3\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$
or $x+3y=0$
So let $y=k$, then
$x=-3k$
So, $X=\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}-3k\\ k\end{array}\right]\sim \left[\begin{array}{c}3\\ -1\end{array}\right]$