First slide

Introduction to Determinants

Question

If $f (x) = |\begin{array}{ccc} 3 & 3 x & 3 x^{2} + 2 a^{2} \\ 3 x & 3 x^{2} + 2 a^{2} & 3 x^{3} + 6 a^{2} x \\ 3 x^{2} + 2 a^{2} & 3 x^{3} + 6 a^{2} x & 3 x^{4} + 12 a^{2} x^{2} + 2 a^{4} \end{array}|$ , then

Moderate

A

f'(x) = 0
B

y = f(x) is a straight line parallel to x-axis
C

$\int_{0}^{2} f (x) dx = 32 a^{4}$
D

none of these

Solution

Applying $C_{3} \to C_{3} - {xC}_{2}, C_{2} \to C_{2} - {xC}_{1}$ , we obtain
$Δ (x) = |\begin{array}{ccc} 3 & 0 & 2 a^{2} \\ 3 x & 2 a^{2} & 4 a^{2} x \\ 3 x^{2} + 2 a^{2} & 4 a^{2} x & 6 a^{2} x^{2} + 2 a^{4} \end{array}|$
$= 4 a^{4} |\begin{array}{ccc} 3 & 0 & 1 \\ 3 x & 1 & 2 x \\ 3 x^{2} + 2 a^{2} & 2 x & 3 x^{2} + a^{2} \end{array}|$
Applying $C_{3} \to C_{3} - {xC}_{2}$ , we get
$Δ (x) = 4 a^{4} |\begin{array}{ccc} 3 & 0 & 1 \\ 3 x & 1 & x \\ 3 x^{2} + 2 a^{2} & 2 x & x^{2} + 2 a^{2} \end{array}|$
Applying $C_{1} \to C_{1} - 3 C_{3}$ , we get
$Δ (x) = 4 a^{4} |\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 1 & x \\ - 4 a^{2} & 2 x & x^{2} + 2 a^{2} \end{array}| = 16 a^{6}$

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