What's the trick to find the real numbers $ \lambda $ for which the following equation system has a nontrivial solution ?
$x_1 + x_5 = \lambda x_1 $
$x_1 + x_3 = \lambda x_2 $
$x_2 + x_4 = \lambda x_3 $
$x_3 + x_5 = \lambda x_4 $
$x_1 + x_4 = \lambda x_5 $
Best Answer
The system can be written in matrix form as $Ax=0$, where $$ A= \begin{bmatrix} 1-\lambda & 0 & 0 & 0 & 1 \\ 1 & -\lambda & 1 & 0 & 0 \\ 0 & 1 & -\lambda & 1 & 0 \\ 0 & 0 & 1 & -\lambda & 1 \\ 1 & 0 & 0 & 1 & -\lambda \end{bmatrix} $$ A homogeneous system has a non trivial solution if and only if the determinant of the matrix is $0$. Developing the determinant with respect to the first row we get $$ \det A= (1-\lambda)\det \begin{bmatrix} -\lambda & 1 & 0 & 0 \\ 1 & -\lambda & 1 & 0 \\ 0 & 1 & -\lambda & 1 \\ 0 & 0 & 1 & -\lambda \end{bmatrix} +\det \begin{bmatrix} 1 & -\lambda & 1 & 0 \\ 0 & 1 & -\lambda & 1 \\ 0 & 0 & 1 & -\lambda \\ 1 & 0 & 0 & 1 \end{bmatrix} $$ Continue the development; you'll find a fifth degree polynomial in $\lambda$, the roots of which answer your question.