First slide

Introduction to Determinants

Question

Let $Δ_{1} = |\begin{array}{ccc} a x & b y & c z \\ x^{2} & y^{2} & z^{2} \\ 1 & 1 & 1 \end{array}|$ and $Δ_{2} = |\begin{array}{ccc} a & b & c \\ x & y & z \\ y z & z x & x y \end{array}|$ then , $∆_{1} - ∆_{2}$ equal

Moderate

A

$(x - 1) (y - 1) (z - 1)$
B

$(x - y) (y - z) (z - x)$
C

$a b c (x - y) (y - z) (z - x)$
D

0

Solution

Multiplying $R_{3}$ of $∆_{1}$ by $x y z,$ we get
$Δ_{1} = \frac{1}{x y z} |\begin{array}{ccc} a x & b y & c z \\ x^{2} & y^{2} & z^{2} \\ x y z & x y z & x y z \end{array}| = \frac{x y z}{x y z} |\begin{array}{ccc} a & b & c \\ x & y & z \\ y z & z x & x y \end{array}| = Δ_{2}$
[taking $x, y, z$ common from $C_{1}, C_{2}, C_{3}$ respectively]
$Δ_{1} - Δ_{2} = 0$

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