First slide

Introduction to Determinants

Question

Suppose $a, b, c$ and $x$ are real numbers. Let $Δ = |\begin{array}{ccc} 1 + a & 1 + a x & 1 + a x^{2} \\ 1 + b & 1 + b x & 1 + b x^{2} \\ 1 + r & 1 + r x & 1 + r x^{2} \end{array}|$ .Then $∆$ is independent of

Moderate

A

$a, b, c$
B

$x$
C

$a, b, c, x$
D

none of these

Solution

Write $Δ = Δ_{1} + Δ_{2},$ where
$Δ_{1} = |\begin{array}{ccc} 1 & 1 + a x & 1 + a x^{2} \\ 1 & 1 + b x & 1 + b x^{2} \\ 1 & 1 + c x & 1 + c x^{2} \end{array}|$
and $Δ_{2} = |\begin{array}{lll} a a x a x^{2} \\ b b x b x^{2} \\ c c x c x^{2} \end{array}| = 0$
$[∵ C_{1} and C_{2} are propotional]$
In $Δ_{1},$ use $C_{2} \to C_{2} - C_{1}, C_{3} \to C_{3} - C_{1}$ to obtain
$Δ_{1} = |\begin{array}{ccc} 1 & a x & a x^{2} \\ 1 & b x & b x^{2} \\ 1 & c x & c x^{2} \end{array}| = 0$
$[∵ C_{2} and C_{3} are propotional]$
Thus $Δ = 0$ and hence independence $a, b, c, x .$

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