Question

Find the value of $\lambda$ for which the system of equations
$(1+\lambda )x_1+x_2+x_3=1$
$x_1+(1+\lambda )x_2+x_3=1$
$x_1+x_2+(1+\lambda )x_3=1$ has infinitely many solutions.

Answer 1

$\left[\begin{array}{ccc}1+\lambda & 1& 1\\ 1& 1+\lambda & 1\\ 1& 1& 1+\lambda \end{array}\right]$

Infinite solution

$\Rightarrow △=0$

$R_1\to R_1-R_3$

$\left[\begin{array}{ccc}\lambda & 1& 1\\ 1& 1+\lambda & 1\\ 1& 1& 1+\lambda \end{array}\right]$

$\lambda \left[\begin{array}{ccc}1& 0& -1\\ 1& 1+\lambda & 1\\ 1& 1& 1+\lambda \end{array}\right]$

$\lambda [{\lambda }^2+2\lambda -1(1-1-\lambda )]$

$=\lambda [{\lambda }^2+2\lambda +\lambda ]$

${\lambda }^2(\lambda +3)$

$\lambda =0$ and $-3$