Spectral radius greater than 1

Question

The eigenvalues of the matrix $$A=\begin{pmatrix}0 & 1/2 & -1 \\ -1/2 & 0 & 1 \\ -1 & -1 & 0\end{pmatrix}$$ are relatively complicated (roots of the polynomial $x\mapsto x^3+x/4+1$). Is there an easy way to see that the spectral radius of this matrix satisfies $\rho(A)>1$, without explicitly calculating the eigenvalues?

Answer 1

Suppose $\rho(A)\le1$ so $|\lambda_i|\le1$ for all $i$. But the product of eigenvalues is $-1$ i.e. $|\lambda_1||\lambda_2||\lambda_3|=1$. This means that all eigenvalues have unit modulus, and in particular the one real eigenvalue is either $1$ or $-1$. Can you check that $-1,1$ is not an eigenvalue?