Q. The value of the determinant $∣ ∣ b + c c + a a + b q + r r + p p + q y + z z + x x + y ∣ ∣ is$

263 136 Determinants Report Error

A

$2 ∣ ∣ a b c p q r x y z ∣ ∣$

B

$∣ ∣ a b c p q r x y z ∣ ∣$

C

$- ∣ ∣ a b c p q r x y z ∣ ∣$

D

0

Solution:

Here, $∣ ∣ b + c c + a a + b q + r r + p p + q y + z z + x x + y ∣ ∣$
$= ∣ ∣ b + c q + r y + z c + a r + p z + x a + b p + q x + y ∣ ∣$
(interchange row and column)
$= ∣ ∣ b + c q + r y + z c + a r + p z + x - 2 c - 2 r - 2 z ∣ ∣$
using $C_{3} \to C_{3} - (C_{1} + C_{2})$
$= - 2 ∣ ∣ b + c q + r y + z c + a r + p z + x c r z ∣ ∣$
(taking -2 common from $C_{3}$ )
$= - 2 ∣ ∣ b q y a p x c r z ∣ ∣$
(using $C_{1} \to C_{1} - C_{3}$ and $C_{2} \to C_{2} - C_{3}$ )
$= 2 ∣ ∣ a p x b q y c r z ∣ ∣$ (using $C_{1} \leftrightarrow C_{2}$ )
$= 2 ∣ ∣ a b c p q r x y z ∣ ∣$
(interchange column and row)