MathNet 2011 Question 80

The matrix $A \in M_k(\mathbb{R})$ ($k \ge 2$) has the property: for every positive integer $n$, $\det(n^2 I_k - A^2) \ge n^k (\det(nI_k - A) + \det(nI_k + A) - 1)$. Prove that $\operatorname{tr}(A) = 0$.