Question

A 2 x 2 matrix M has eigen values 2 & 3 with eigen vectors $\left[\begin{array}{c}2\\ 1\end{array}\right]\&\left[\begin{array}{c}1\\ 2\end{array}\right]$ respectively. The sum of all elements of Matrix M is _______.

• A. 5

Answer 1

The correct option is A 5
Let matrix $M=\left[\begin{array}{cc}p& q\\ r& s\end{array}\right]$

Now for eigen value ${\lambda }_1$ = 2 & eigen vector $X_1=\left[\begin{array}{c}2\\ 1\end{array}\right]$

$M{X}_1={\lambda }_1{X}_1$

$\Rightarrow \left[\begin{array}{cc}p& q\\ r& s\end{array}\right]\left[\begin{array}{c}2\\ 1\end{array}\right]=2\left[\begin{array}{c}2\\ 1\end{array}\right]$

$\Rightarrow \left[\begin{array}{c}2p+q\\ 2r+s\end{array}\right]=\left[\begin{array}{c}2\\ 4\end{array}\right]$ ..... (1)

Also for eigen value ${\lambda }_2$ with eigen vector $X_2$

$M{X}_2={\lambda }_2{X}_2$

$\left[\begin{array}{cc}p& q\\ r& s\end{array}\right]\left[\begin{array}{c}1\\ 2\end{array}\right]=3\left[\begin{array}{c}1\\ 2\end{array}\right]$

$\left[\begin{array}{c}P+2q\\ r+2s\end{array}\right]=\left[\begin{array}{c}3\\ 6\end{array}\right]$ ...(2)

From (1) & (2) we have,

2p + q = 4; 2r + s = 2
p + 2q = 3 ; r + 2s = 6
$\Rightarrow$ 3p + 3q =7
& 3r + 3s = 8
$\Rightarrow 3(p+q+r+s)=15$
$\Rightarrow$ p + q + r + s = 5