In a homogeneous system, if there exists a non-trivial solution, does that mean there is no trivial solution

Question

In a homogeneous system, if there exists a non-trivial solution, does that mean there is no trivial solution?

I am having a hard time coming up with the answer for this. I'm thinking that all homogeneous systems can have a trivial solution, but there can be cases when there exists a non-trivial solution as well. I learned this just now though, so I may be extremely wrong.

Thanks

Answer 1

A homogeneous linear system always has the trivial solution, no matter what coefficients it has. It may, however, have no non-trivial solutions; if so, any linear system with the same coefficient matrix as the homogeneous system has exactly one solution.