For any real matrix $A$ of dimension $3$ and for any $n \geqslant 2$, show that$$\det(A-I_3)^2+ \det(A-2I_3)^2 +\cdots+\det(A-2nI_3)^2\geqslant\frac{\det A^2}{\binom{4n}{2n}-1}.$$
I know that $\det(A-xI_3)$ is equal to zero polynomial from Cayley-Hamilton, but I do not know what I should use to deal with the binomial and to solve it. Any help please? Thanks in advance.