That is the same (1- k)x+ y= 0 and -x+ (1- k)y= 0. One obvious solution is x=y= 0 but that is the "trivial solution".

You would get that, for example, if you solved the first equation for y: y= (k- 1)x. Putting that into the second equation, we get (-1+ k(k-1))x= k(k-1)x so that, if x is not 0, we can divide on both sides to get (-1+ k(k-1))= k(k-1).