Question

A matrix has eigen values $-1$ and $-2$. The corresponding eigen vectors are $\left[\begin{array}{c}1\\ -1\end{array}\right]$ and $\left[\begin{array}{c}1\\ -2\end{array}\right]$ respectively. The matrix is

• A. $\left[\begin{array}{cc}1& 1\\ -1& -2\end{array}\right]$
• B. $\left[\begin{array}{cc}1& 2\\ -2& -4\end{array}\right]$
• C. $\left[\begin{array}{cc}1& 0\\ 0& -2\end{array}\right]$
• D. $\left[\begin{array}{cc}0& 1\\ -2& -3\end{array}\right]$

Answer 1

The correct option is D $\left[\begin{array}{cc}0& 1\\ -2& -3\end{array}\right]$

${\lambda }_1=-1,X_1=\left[\begin{array}{c}1\\ -1\end{array}\right]$
${\lambda }_2=-2,X_2=\left[\begin{array}{c}1\\ -2\end{array}\right]$
Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
$AX=\lambda X$
$\Rightarrow \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}1\\ -1\end{array}\right]=-1\left[\begin{array}{c}1\\ -1\end{array}\right]$

$\begin{array}{c}a-b=-1\\ c-d=1\end{array}\}$ ......(1)
and $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}1\\ -2\end{array}\right]=-2\left[\begin{array}{c}1\\ -2\end{array}\right]$

$\begin{array}{c}a-2b=-2\\ c-2d=4\end{array}\}$ ......(2)

From (1) and (2)

$a=0,b=1,c=-2,d=-3$

$\therefore$ $A=\left[\begin{array}{cc}0& 1\\ -2& -3\end{array}\right]$