First slide

System of linear equations

Question

If the system of equations $x = c y + b z, y = a z + c x, z = b x + a y$ has a nonzero solution then $a^{2} + b^{2} + c^{2} + 2 a b c =$

Moderate

A

0
B

1
C

2
D

5

Solution

$x = c y + b z, y = a z + c x, z = b x + a y \Rightarrow [\begin{matrix} - 1 & c & b \\ c & - 1 & a \\ b & a & - 1 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$
$Given equations has non zero solution \Rightarrow |\begin{array}{ccc} - 1 & c & b \\ c & - 1 & a \\ b & a & - 1 \end{array}| = 0$
$\begin{array}{l} \Rightarrow - 1 (1 - a^{2}) - c (- 1 - a b) + b (a c + b) = 0 \Rightarrow - 1 + a^{2} + c^{2} + a b c + a b c + b^{2} = 0 \\ \Rightarrow a^{2} + b^{2} + c^{2} + 2 a b c = 1 \end{array}$

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