We were given a system of equations: $$\lambda x+y+z=p$$ $$\lambda y+z+x=q$$ $$\lambda z+x+y=r$$ We are required to find the relations between $p,q,r,\lambda$ for the case when no solution exists for the given system.
So I created the augmented matrix: $$ \left[ \begin{array}{ccc|c} \lambda&1&1&p\\ 1&\lambda&1&q\\ 1&1&\lambda&r \end{array} \right] $$
Converting it to row-echelon form we get: $$ \left[ \begin{array}{ccc|c} \lambda&1&1&p\\ 0&\frac{\lambda^2-1}{\lambda}&\frac{\lambda-1}{\lambda}&q-\frac{p}{\lambda}\\ 0&0&\frac{\lambda^3-1}{\lambda^2+\lambda}&r+\frac{r-p-q}{\lambda} \end{array} \right] $$
Now for no solution the matrix needs to have a degenerate case ie. one whole row of zeroes and a non-zero value on the RHS. So for this the possible cases are:
- $\lambda=1$ and $p+q\ne 2r$
- $\lambda=1$ and $p\ne q$
But on observation I also found another case that's possible ie. $\lambda=-2$ and $p+q+r\ne0$. How do I arrive at this using Gauss elimination because I can't see it in the augmented matrix?
Any help/alternate solution would be appreciated.