First slide

Introduction to Determinants

Question

If x, y, z are different from zero and $Δ = |\begin{array}{ccc} a & b - y & c - z \\ a - x & b & c - z \\ a - x & b - y & c \end{array}| = 0$ , 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 $R_{2} \to R_{2} - R_{1}, R_{3} \to R_{3} - R_{1}$ , we get
$Δ = |\begin{array}{ccc} a & b - y & c - z \\ - x & y & 0 \\ - x & 0 & z \end{array}| = 0$
Expanding along C₃ we get
$(c - z) |\begin{matrix} - x & y \\ - x & 0 \end{matrix}| + z |\begin{array}{cc} a & b - y \\ - x & y \end{array}| = 0$
or (c - z) (xy) + z(ay + bx - xy) = 0
or cxy - xyz + ayz + bxz - xyz = 0
or ayz + bzx + cxy = 2xyz
or $\frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 2$

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