We know that a homogeneous system of linear equations always has a trivial solution. Also, if there are more variables than equations, we are guaranteed to have a non-trivial solution.
However, if there are more equations than variables, is there a way to tell if there exists a nontrivial solution? (I am interested in solving a system of linear equations modulo integer $n$ ). If we represent such linear systems with a matrix, we know that the rows of the matrix must be linearly dependent. However, this condition is not sufficient for having a non-trivial solution.
Any hint/insight would be really appreciated.