How to prove if the statement "If A is a nonsingular matrix, then the homogeneous system Ax = 0 has a nontrivial solution" is true or false? Statement: If  is a nonsingular matrix, then the homogeneous system  = 0 has a nontrivial solution
We know that if A is an n × n non–singular
matrix, then the homogeneous system
AX = 0 has only the trivial solution X = 0.
Hence if the system AX = 0 has a non–trivial
solution, A is singular.
Example:

By solving the row echelon form of A, we get:

Because of this, we can say that A is singular because we got its reduced row echelon form,
and consequently AX = 0 has a non–trivial
solution x = −1, y = −1, z = 1
More generally, if A is
row–equivalent to a matrix containing a zero
row, then A is singular. For then the
homogeneous system AX = 0 has a
non–trivial solution.
Now, my issue here is that I hesitate to conclude if the given statement above is considered true or false because of the presence of the possibility in the matrix that it can be either trivial or non-trivial. I may want to know what is the final verdict for the statement above if it's true or false.
Your responses would be highly appreciated as this would help me a lot to get a clearer context.
Thank you very much!

If $A$ is nonsingular, it has an inverse $A^{-1}.$ (If you have not seen this fact or do not want to use it, I'll remove this answer.) Therefore if $x$ is any solution to $Ax=0$, then $$x = (A^{-1}A)x = A^{-1}(Ax) = A^{-1}0 =0,$$
so that $Ax=0$ indeed only has the trivial solution.
