$\displaystyle x+2y+2z=pz$
$\displaystyle 2x+y+2z=py$
$\displaystyle 2x+2y+z=pz$
For what values of p, these equations have non-trivial solution.
How to solve this question. And also tell me what is meant by non-trivial solution
These can all be written in the form
$\displaystyle x+ 2y+ (2- p)z= 0$
$\displaystyle 2x+ (1- p)y+ 2z= 0$
$\displaystyle 2x+ 2y+ (1- p)z= 0$
which obviously has the solution x= y= z= 0. That is the "trivial" solution. If the coefficient matrix $\displaystyle \begin{bmatrix}1 & 2 & 2- p \\ 2 & 1- p & 2 \\ 2 & 2 & 1- p\end{bmatrix}$
has determinant 0 (is not invertible) then there will also exist other "non-trivial" solutions.