# trivial and non-trivial solution of homogeneous equation [closed]

Suppose I have system of 3 equations $$a_1x+b_1y+c_1z=0$$ $$a_2x+b_2y+c_2z=0$$ $$a_3x+b_3y+c_3z=0$$ and cofficient matrix $A=$$\begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{pmatrix}$$$ So I have been told that solution of this matrix will be non-trivial if $|A|=0$ and trivial in any other case. As far as I know non trivial solution means solutions is not equal to zero but in any case $x,y,z=0$ will satisfy given equations regardless of it's value of determinant. So, why do we call it "non-trivial" solution?

## closed as unclear what you're asking by Alex M., Behrouz Maleki, Claude Leibovici, Daniel W. Farlow, Stefan PerkoJan 5 '17 at 15:48

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• Since the zero solution is the "obvious" solution, hence it is called a trivial solution. Any solution which has at least one component non-zero (thereby making it a non-obvious solution) is termed as a "non-trivial" solution. – Anurag A Aug 13 '15 at 17:26
• If determinant is zero, then apart from trivial solution there will be infinite number of other, non-trivial, solutions. – Kaster Aug 13 '15 at 17:33
• Precisely because they are different from the trivial solution. – Yves Daoust May 17 '16 at 7:23

If $x=y=z=0$ then trivial solution And if $|A|=0$ then non trivial solution that is the determinant of the coefficients of $x,y,z$ must be equal to zero for the existence of non trivial solution. Simply if we look upon this from mathwords.com
For example, the equation $x + 5y = 0$ has the trivial solution $x = 0, y = 0.$ Nontrivial solutions include $x = 5, y = –1$ and $x = –2, y = 0.4.$
• The OP knows this. He asks why, in the case $\det A \ne 0$, the null solution is called "non-trivial", given that it is trivial. Well, the question is non-sensical and it is rooted in the OP's misuse of a mathematical term. – Alex M. Jan 2 '17 at 15:59