SEEMOUS 2014 Problem 3

Let $A\in M_n(\mathbb{C})$ and $a\in \mathbb{C}$ such that $A-A^*=2aI_n$ , where $A^*=(\overline{A})^T$ and $I_n$ is identity matrix. (i) Show that $|\det A|\ge |a|^n$ .

(ii) Show that if $|\det A|=|a|^n$ then $A=aI_n$ .