Home » Uncategorized » GATE 2015 EE - SET 1 - Question 3

# GATE 2015 EE - SET 1 - Question 3

Solution : Let A be a 2 x 2 matrix with the sum of diagonal elements as - 6.

Let $\lambda _{1}$ and $\lambda _{2}$ be the eigen values of A.

$\therefore$ The sum of the diagonal elements of A = - 6 .

$\Rightarrow$  $\lambda _{1} + \lambda _{2}$ = -6 -----------------(1)

Det of A = $\left | A \right |$ = $\lambda _{1}$ $\lambda _{2}$

Now we have to find the maximum value of  $\lambda _{1}$ and $\lambda _{2}$

Let f = $\lambda _{1}$ and $\lambda _{2}$

$\therefore$ $\lambda _{1} (-6- \lambda _{1} )$ (from (1))

$\therefore$ f = - 6 $\lambda _{1}$ - $\lambda {_{1}}^{2}$

$\Rightarrow$ f' = -6 - 2 $\lambda _{1}$

For f to have  maximum , f ' = 0

$\Rightarrow$ - 6 - 2 $\lambda _{1}$ = 0

$\lambda _{1}$  = - 3

Now f '' = - 2 < 0

$\therefore$ f has a maximum at $\lambda _{1}$  = - 3

From (1)  $\lambda _{1}$  + $\lambda _{2}$ = -6

$\Rightarrow$  - 3 + $\lambda _{2}$ = -6

$\Rightarrow$  - 3 + $\lambda _{2}$ = -6

$\Rightarrow$   $\lambda _{2}$ = -3 .

The maximum value of the determinant of  A = $\lambda _{1}$ $\lambda _{2}$

= (- 3 ) x (-3) = 9.

Hence , the correct answer is 9.

Translate »