Question 1055240
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The determinant of the matrix is 

{{{1 +2*alpha*beta - 1 - (alpha)^2 - (beta)^2}}} = {{{-(alpha - beta)^2}}}.


Therefore, the given system has a non-trivial solution if and only if 

{{{alpha - beta}}} = 0,  or,  equivalently,  {{{alpha}}} = {{{beta}}}.
</pre>

Honestly, I don't know what condition on {{{alpha}}} and {{{beta}}} is written in your post (or what you were going to write there).


Unfortunately.