Prove if the statement “If A is a nonsingular matrix, then the homogeneous system Ax = 0 has a nontrivial solution” is true or false

Question

Statement: If 𝐴 is a nonsingular matrix, then the homogeneous system 𝐴𝑥 = 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!

Answer 1

If $A$ is a square invertible matrix, being full rank and being invertible is equivalent.

Therefore we can use the rank–nullity theorem and see that the kernel of your matrix is a vector space whose dimension is necessarily $0$, meaning that it only contains the null vector.

So there is only the trivial solution.