First slide

Introduction to Determinants

Question

If $Δ (x) = |\begin{array}{ccc} 1 & x & x + 1 \\ 2 x & x (x - 1) & x (x + 1) \\ 3 x (x - 1) & x (x - 1) (x - 2) & x (x^{2} - 1) \end{array}|$ then $∆ (100)$ equals

Moderate

A

0
B

-100
C

100
D

-100!

Solution

Taking x common from $C_{2}, x + 1$ from $C_{3}$ and x – 1 from $R_{3}$ , we get
$Δ (x) = x (x + 1) (x - 1) |\begin{array}{ccc} 1 & 1 & 1 \\ 2 x & x - 1 & x \\ 3 x & x - 2 & x \end{array}|$
Applying $C_{1} \to C_{1} - C_{3}, C_{2} \to C_{2} - C_{3}$ we get
$Δ (x) = x (x + 1) (x - 1) |\begin{array}{ccc} 0 & 0 & 1 \\ x & - 1 & x \\ 2 x & 2 & x \end{array}| = 0$
[ $C_{1} a n d C_{2}$ are proportional]
Thus , $∆ (100) = 0$

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