First slide

Introduction to Determinants

Question

If $f (x) = |\begin{array}{lll} x a a \\ a x a \\ a a x \end{array}| = 0$ , then which of the following is correct ?

Moderate

A

f'(x) = 0 and f"(x) = 0 has one common root
B

f(x) = 0 and f'(x) = 0 has one common root
C

sum of roots of f(x) = 0 is -3a
D

none of these

Solution

Applying $C_{1} \to C_{1} + C_{2} + C_{3}$ , we get
$Δ = |\begin{array}{lll} x + 2 a a a \\ x + 2 a x a \\ x + 2 a a x \end{array}| = (x + 2 a) |\begin{array}{ccc} 1 & a & a \\ 1 & x & a \\ 1 & a & x \end{array}|$
Applying $R_{1} \to R_{1} - R_{2} and R_{2} \to R_{2} - R_{3}$ , we get
$Δ = (x + 2 a) |\begin{array}{ccc} 0 & a - x & 0 \\ 0 & x - a & a - x \\ 1 & a & x \end{array}| = (x - a)^{2} (x + 2 a) r o o t s a r e a, a, - 2 a$

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