Q. If $Δ = ∣ ∣ - x b a a - x b b a - x ∣ ∣$ , then a factor of $Δ$ is

1216 238 Determinants Report Error

A

$a + b + x$

B

$x^{2} - (a - b) x + a^{2} + b^{2} + ab$

C

$x^{2} + (a + b) x + a^{2} + b^{2} - ab$

D

$x$

Solution:

Applying $C_{1} \to C_{1} + C_{2} + C_{3}$ we get
$Δ = ∣ ∣ a + b - x a + b - x a + b - x a - x b b b - x ∣ ∣ = (a + b - x) ∣ ∣ 111 a - x b b a - x ∣ ∣$
$= (a + b - x) ∣ ∣ 100 a - x - a b - a b a - b - x - b ∣ ∣$
[using $R_{2} \to R_{2} - R_{1}$ and
$R_{3} \to R_{3} - R_{1}]$
$= (a + b - x) [(x + a) (x + b) + (a - b)^{2}]$ [expanding along $C_{1}]$
$= (a + b - x) [x^{2} + (a + b) x + a^{2} + b^{2} - ab]$