A homogeneous system $Ax = 0$ with $\det(A) = 0$?

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Suppose we have $Ax = 0$ with a square matrix $A$. I tried to prove the following:

$\det(A) = 0$ iff the system $Ax = 0$ is inconsistent.

I am not sure if I misunderstood the consistency definition. My intuition is: If at least there is one solution, then the system is consistent. If there is NO solution and if one can derive a contradiction from equations, then the system is inconsistent. In the case of $Ax = 0$, there is always a solution $x = \mathbf{0}$. It means that system is always consistent. Then, I suppose I have to simply disprove this argument.

But I am not sure about that, so I am here.

Thanks in advance!

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1 Answer

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You're correct. I suspect it should read $\det(A)=0$ if and only if the system $Ax=0$ is not independent (has more than one solution).

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