$\displaystyle x+2y+2z=pz$

$\displaystyle 2x+y+2z=py$

$\displaystyle 2x+2y+z=pz$

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

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- Nov 17th 2010, 02:13 AMkumaran5555Non-trivial solution
$\displaystyle x+2y+2z=pz$

$\displaystyle 2x+y+2z=py$

$\displaystyle 2x+2y+z=pz$

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 - Nov 17th 2010, 02:18 AMHallsofIvy
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.