Eliminate $l, m, n$ from the equations
$a l + c m + b n = 0, c l + b m + a n = 0. b l + a m + c n = 0$ .
and express the result in the simplest form.

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Solution

$a l + c m + b n = 0 c l + b m + a n = 0 b l + a m + c n = 0$
$⎡ ⎢ ⎣ \begin{matrix} a & c & b c & b & a b & a & c \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} l m n \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 000 \end{matrix} ⎤ ⎥ ⎦$
This can be eliminated as
$∣ ∣ ∣ ∣ \begin{matrix} a & c & b c & b & a b & a & c \end{matrix} ∣ ∣ ∣ ∣ = 0 \Rightarrow R_{1} = R_{1} + R_{2} + R_{3} ∣ ∣ ∣ ∣ \begin{matrix} a + b + c & a + b + c & a + b + c c & b & a b & a & c \end{matrix} ∣ ∣ ∣ ∣ = 0 (a + b + c) ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 c & b & a b & a & c \end{matrix} ∣ ∣ ∣ ∣ = 0 \Rightarrow (a + b + c) {1 (b c - a^{2}) - 1 (c^{2} - a b) + 1 (a c - b^{2})} = 0 \Rightarrow (a + b + c) (b c + a b + a c - a^{2} - b^{2} - c^{2}) = 0$

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