First slide

System of linear equations

Question

If the system of equations $a x + y + z = 0, x + b y + z = 0, x + y + c z = 0, (a, b, c \neq 1)$ has a non trivial solution ( non-zero solution) , then $\frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c} =$

Moderate

A

1
B

-1
C

0
D

None

Solution

Given system of equations has a nontrivial solution $\Rightarrow^{}$ coefficient matrix is singular
$\Rightarrow |\begin{matrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{matrix}| = \Rightarrow a (b c - 1) - 1 (c - 1) + 1 (1 - b) = 0 \Rightarrow a b c - a - c + 1 + 1 - b = 0$
$\begin{array}{l} \Rightarrow a b c = a + b + c - 2 \\ \frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c} = \frac{(1 - b) (1 - c) + (1 - a) (1 - c) + (1 - a) (1 - b)}{(1 - a) (1 - b) (1 - c)} \\ \Rightarrow \frac{3 - 2 (a + b + c) + b c + a c + a b}{1 - (a + b + c) + a b + b c + c a - a b c} = \frac{3 - 2 (a + b + c) + b c + c a + a b}{1 - (a + b + c) + a b + b c + c a - (a + b + c - 2)} = 1 \end{array}$

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