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: 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?