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?

Question

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!

Answer 1

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.

Answer 2

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.

Answer 3

The given statement is FALSE. Since $A$ is non-singular means $det(A)\ne 0$ hence the homogeneous system $Ax=0$ can be solved as :
$A^{-1}(Ax)=A^{-1}0$
$\implies (A^{-1}A)x=0$
$\implies Ix=0$
$\implies x=0$
i.e. Trivial solution