[Math] If $A$ is a square matrix and $Ax = b$ has a unique solution for some $b$, is $A$ necessarily invertible

Question

Let $A$ be a square matrix. Suppose that $A x = b$ has a unique solution for some $b$. Is $A$ necessarily invertible?

I said no because the invertible matrix theorem states that $A x = b$ has a unique solution for each $b$. Is this correct or does the wording not make a difference?

Answer 1

Yes. Let $x$ be the unique solution of $Ax=b$. If $y$ is a nonzero solution of $Ay=0$, then $A(y+x)=b$ and $y+x\neq x$, a contradiction.