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Determine all the values of c for which the system has nontrivial solutions, and then find all the solutions.

x + 2y + cz = 0

3x - y = 0

-2x + y + z = 0

OK, the system has nontrivial solutions if the determinant = 0.

So I set the determinant = 0 and solve for c.

Using cofactor expansion, I have:

c(3 + 2) + (-1 - 6) ==> c = 7/5.

I plug this into the system and using elementary row operations I still only come up with only the trivial solution x = 0, y = 0,

z = 0. I can't go through the steps here. It would be way too tedious, but I've gone over it several times and come up with the same result. What gives? I was promised that if the det = 0 the system has either 0 solutions or infinitely many solutions. Since x,y,z = 0 is a solution there must be infinitely many solutions.

Where have I gone wrong?

Thanks.