Question

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

$A=\left[\begin{array}{cc}P& 1\\ 1& 2\end{array}\right]$

Let ${\lambda }_1$ and ${\lambda }_2$ be the eigen values of matrix A

For P = 2, $\frac{{\lambda }_1}{{\lambda }_2}=\frac{3}{1}$ (given)

$\Rightarrow {\lambda }_1=3{\lambda }_2$

Sum of eigen values $=({\lambda }_1+{\lambda }_2)=(2+P)...\left(i\right)$

Product of eigen values $={\lambda }_1{\lambda }_2=(2P-1)...\left(ii\right)$

Putting ${\lambda }_1=3{\lambda }_2$ in equation (i) and equation (ii), we get

$4{\lambda }_2=(2+P)\Rightarrow {\lambda }_2=\frac{(2+P)}{4}$

And $3{\lambda }_2^2=2P-1$

$\Rightarrow 3\times {\left(\frac{P+2}{4}\right)}^2=2P-1$

On solving the above equation, we get

$P=2,\frac{14}{3}$

$\therefore$ The another value of P = $\frac{14}{3}$