Question

The two eigen value of the matrix $\left[\begin{array}{cc}2& 1\\ 1& P\end{array}\right]$ have ratio of 3:1, for P =2. What another value of P for which the Eigen values have the same ratio of 3:1

• A. -2
• B. 1
• C. 7/3
• D. 14/3

Answer 1

The correct option is D 14/3

$A=\left[\begin{array}{cc}2& 1\\ 1& P\end{array}\right]$
Let ${\lambda }_1$ and ${\lambda }_2$ be the eigen value of matrix A.

$\therefore$ $\frac{{\lambda }_1}{{\lambda }_2}=\frac{3}{1}forP=2$
Sum of eigen value $={\lambda }_1+{\lambda }_2=2+P$ ...(1)
product of eigen values $={\lambda }_1{\lambda }_2=2P-1$ ...(2)
$\frac{{\lambda }_1}{{\lambda }_2}=\frac{3}{1}$
$\Rightarrow {\lambda }_1=3{\lambda }_2$
(1) $\Rightarrow 3{\lambda }_2+{\lambda }_2=2+P$
$4{\lambda }_2=2+P$
${\lambda }_2=\frac{2+P}{4}$
(2) $\Rightarrow$ $3{\lambda }_2^2=2P-1$
$\Rightarrow$ $3(\frac{P+2}{4}{)}^2=2P-1$
$\Rightarrow$ $P=2,\frac{14}{3}$

Alternative Solution :
$A=\left[\begin{array}{cc}2& 1\\ 1& P\end{array}\right]$
Characteristic equation is :
$|A-I\lambda |=0$
$\left|\begin{array}{cc}2-\lambda & 1\\ 1& P-\lambda \end{array}\right|=0$
$(2-\lambda )(P-\lambda )-1=0$

${\lambda }^2-(P+2)\lambda +(2P-1)=0$
By putting values of P from options.
Option (D) 14/3 above equations gives value 5,5/3
Henceratio of two eigen values $=\frac{5}{5/3}=\mathrm{3.1.}$