System of linear equations

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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 $a^{2} + b^{2} + c^{2} + 2 a b c =$

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Solution

$T h e c o n d i t i o n f o r a h o m o g e n e o u s s y s t e m o f e q u a t i o n s t o h a v e a n o n - z e r o s o l u t i o n i s Δ = 0$
$\Rightarrow |\begin{matrix} - 1 & c & b \\ c & - 1 & a \\ b & a & - 1 \end{matrix}| = 0 \Rightarrow - 1 (1 - a^{2}) - c (- c - a b) + b (a c + b) = 0$
$\Rightarrow - 1 + a^{2} + c^{2} + a b c + a b c + b^{2} = 0$
$\Rightarrow a^{2} + b^{2} + c^{2} + 2 a b c = 1$

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System of linear equations

Remember concepts with our Masterclasses.

If the system of equations x=cy+bz, y=az+cx, z=bx+ayhas a nonzero solution thena2+b2+c2+2abc=

The condition for a homogeneous system of equations to have a non-zero solution is Δ=0⇒-1cbc-1aba-1=0 ⇒-11-a2-c-c-ab+bac+b=0⇒ −1+a2+c2+abc+abc+b2=0⇒ a2+b2+c2+2abc=1

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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 $a^{2} + b^{2} + c^{2} + 2 a b c =$