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

Solution

$\begin{aligned} Given that Δ = |\begin{array}{ccc} a & b - y & c - z \\ a - x & b & c - z \\ a - x & b - y & c \end{array}| = 0 \\ Applying R_{2} \to R_{2} - R_{1}, R_{3} \to R_{3} - R_{1}, we get \\ Δ = |\begin{array}{ccc} a & b - y & c - z \\ - x & y & 0 \\ - x & 0 & z \end{array}| = 0 \end{aligned}$
$Expanding along C_{3}, we get$
$\begin{array}{l} (c - z) |\begin{array}{ll} - x & y \\ - x & 0 \end{array}| + z |\begin{array}{cc} a & b - y \\ - x & y \end{array}| = 0 \\ \Rightarrow (c - z) (x y) + z (a y + b x - x y) = 0 \\ \Rightarrow c x y - x y z + a y z + b x z - x y z = 0 \\ \Rightarrow a y z + b z x + c x y = 2 x y z \\ \Rightarrow \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 2 \end{array}$

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