First slide

Introduction to Determinants

Question

a, b, c are distinct real numbers, not equal to one. If ax + y + z = 0, x + by + z = 0, and x + y + cz = 0 have a non-trivial solution, then the value of $\frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c}$ is equal to

Moderate

A

-1
B

1
C

zero
D

none of these

Solution

Since the system has non-trivial solution, we have
$|\begin{matrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{matrix}| = 0$
Applying $R_{1} \to R_{1} - R_{2}, R_{2} \to R_{2} - R_{3}$ , we get
$Δ = |\begin{array}{ccc} a - 1 & 1 - b & 0 \\ 0 & b - 1 & 1 - c \\ 1 & 1 & c \end{array}| = 0$
or c(1 - a)(1 - b) + (1 - b)(1 - c) - (1 - c)(a - 1) = 0
Dividing throughout by (1 - a) (1 - b) (1 - c), we get
$\frac{c}{1 - c} + \frac{1}{1 - a} + \frac{1}{1 - b} = 0$
or $- 1 + \frac{1}{1 - c} + \frac{1}{1 - a} + \frac{1}{1 - b} = 0$
or $\frac{1}{1 - c} + \frac{1}{1 - a} + \frac{1}{1 - b} = 1$

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