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I need your help with this tricky problem.

Determine the value of \[\lambda\] so that the following linear equation system of three unknowns has only the zero solution:

\[\begin{pmatrix} 1 &1 &1 &0 \\ 1 &\lambda &1 &0 \\ \lambda &1 &1 &0 \end{pmatrix}\]

I have started working on the system to bring it to the form in which I can answer the question, and after one stage I got here:

\[\begin{pmatrix} 1 &1 &1 &0 \\ 0 &\lambda-1 &0 &0 \\ 0 &1-\lambda &1-\lambda &0 \end{pmatrix}\]

Now here is where I am stuck, I don't know how to proceed, knowing that any division by \[\lambda-1\] or \[1-\lambda\] is not allowed since it can be a division by 0.

Will appreciate your guidance. Thanks.