First slide

Introduction to Determinants

Question

If $x, y, z$ are different from zero and $Δ = |\begin{matrix} a & b - y & c - z \\ a - x & b & c - z \\ a - x & b - y & c \end{matrix}| = 0$ then the value of the expression $\frac{a}{x} + \frac{b}{y} + \frac{c}{z}$ is

Moderate

A

0
B

-1
C

1
D

2

Solution

Applying , ${\bar{R}}_{2} \to {\bar{R}}_{2} - {\bar{R}}_{1}, {\bar{R}}_{3} \to {\bar{R}}_{3} - {\bar{R}}_{1}$ we get
$Δ = |\begin{matrix} a & b - y & c - z \\ - x & y & 0 \\ - x & 0 & z \end{matrix}| = 0$
Expanding along $C_{3}$ , we get
$(c - z) |\begin{matrix} - x & y \\ - x & 0 \end{matrix}| + z |\begin{matrix} a & b - y \\ - x & y \end{matrix}| = 0$
$\Rightarrow (c - z) (x y) + z (a y + b x - x y) = 0$
$\Rightarrow c x y - x y z + a y z + b x z - x y z = 0$
$\Rightarrow a y z + b z x + c x y = 2 x y z$
$\Rightarrow \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 2$

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