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 $A=\begin{equation} \begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{pmatrix} \end{equation}$ 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 Perko Jan 5 '17 at 15:48

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • 5
    $\begingroup$ 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. $\endgroup$ – Anurag A Aug 13 '15 at 17:26
  • 1
    $\begingroup$ If determinant is zero, then apart from trivial solution there will be infinite number of other, non-trivial, solutions. $\endgroup$ – Kaster Aug 13 '15 at 17:33
  • 1
    $\begingroup$ Precisely because they are different from the trivial solution. $\endgroup$ – 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 $x = –2, y = 0.4.$

  • 2
    $\begingroup$ 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. $\endgroup$ – Alex M. Jan 2 '17 at 15:59

Not the answer you're looking for? Browse other questions tagged or ask your own question.