Prove $Ax = \frac{1}{2}x$ only has the trivial solution where $A$ has all integer entries.

Question

Prove $Ax = \frac{1}{2}x$ only has the trivial solution where $A$ is a $n \times n$ matrix with integer entries and $x = (x_1, \ldots , x_n)$.

I am a bit rusty on my linear algebra and trying to review. I tried using the Invertible Matrix theorem. The problem was I couldn't seem to gain any traction with any of the equivalent statements.

Here is the link for anyone that needs a refresher: Invertible Matrix Theorem

Looking for hints rather than a specific solution. 

Answer 1

Hint: The eigenvalues of $A$ are the roots of its characteristic polynomial, which is a monic polynomial with integer coefficients.

Answer 2

Hint: Evaluate the characteristic polynomial of $A$ at $\lambda=\frac12$. Since all coefficients are integers and the leading term is $\lambda^n$, what can you say about the value?

Answer 3

Write it as $x=2Ax$. If this has a nonzero solution, it has a nonzero rational solution and indeed a nonzero integer solution. But if $x$ has integer entries, $x=2Ax$ has even entries. So then $x=2Ax$ has entries divisible by $4$, etc.....