AX = 0
In this case if A is nonsingular. X = 0 is the only solution. It is a trivial solution.
A system AX = 0 of n equations in n unknowns has non trivial solution iff the coefficieint matrix is singular.
To solve the system with singular A in 3 variables (as an example) Put z =k and solve the remaining equations for x and y in terms of k. This is the general solution to the problem.
A homogeneous system of linear equations is always consistent.