First slide

Introduction to Determinants

Question

If $Δ = |\begin{array}{ccc} - x & a & b \\ b & - x & a \\ a & b & - x \end{array}|$ , then a factor of $∆$ is

Moderate

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

a + b - x

Solution

Applying $C_{1} \to C_{1} + C_{2} + C_{3}$ , we get
$\begin{matrix} Δ = |\begin{array}{ccc} a + b - x & a & b \\ a + b - x & - x & a \\ a + b - x & b & - x \end{array}| = (a + b - x) |\begin{array}{ccc} 1 & a & b \\ 1 & - x & a \\ 1 & b & - x \end{array}| \\ = (a + b - x) |\begin{array}{ccc} 1 & a & b \\ 0 & - x - a & a - b \\ 0 & b - a & - x - b \end{array}| [Applying R_{2} \to R_{2} - R_{1} and R_{3} \to R_{3} - R_{1}] \\ = (a + b - x) [(x + a) (x + b) + (a - b)^{2}] [expanding along C_{1}] \\ = (a + b - x) [x^{2} + (a + b) x + a^{2} + b^{2} - ab] \end{matrix}$

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