Non zero solution of homogenous system equations

Question

If the system of linear equations $$x+2ay+az=0$$ $$x+3by+bz=0$$ $$x+4cy+cz=0$$ has a non-zero solution, then find a relation between $a, b, c$ .

My attempt

I tried to find the discriminant to be zero assuming this to be a non trivial solution but it came out wrong.

Only if the determinant of the homogeneous system is equal zero, we will find other solutions than the zero vector. — Matko, Dec 23 '18 at 11:14
I have found the following relation between $a,b,c$ : $bc-2ac+ab=0$ . — Matko, Dec 23 '18 at 11:16
I have found the same but it doesn't match the options. Maybe the solution is wrong — Aladdin, Dec 23 '18 at 11:23
The determinant is just $a(2c-b)-bc$. From here it's the condition that @Matko found: $ab+bc=2ac$. — Michael Hoppe, Dec 23 '18 at 18:10

score 1 · Accepted Answer · answered Dec 23 '18 at 11:02

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Hint: Eliminating $x$ we get $$y(3b-2a)+z(b-a)=0$$ $$y(4c-2a)+z(c-a)=0$$ Now you have distinguish several cases.

answered Dec 23 '18 at 11:02

Dr. Sonnhard Graubner

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By distinguishing several cases, what do you mean – Aladdin Dec 23 '18 at 11:26
I mean $$b-a\neq 0$$ or $$b-a=0$$ etc – Dr. Sonnhard Graubner Dec 23 '18 at 11:28

score 0 · Answer 2 · answered Dec 23 '18 at 18:23

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As the determinant is $ab+bc-2ac$ we have a non-trivial kernel iff $b(a+c)=2ac$. Now if $a=-c$ it follows that $a=c=0$ and $b$ may be any real number. Otherwise $b=\frac{2ac}{a+c}$.

answered Dec 23 '18 at 18:23

Michael Hoppe

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Non zero solution of homogenous system equations

2 Answers2