Nonnegative Matrix

Question

Let $A=E+\sqrt{-1}B$, where $E=diag\{0,1,\cdots,1\}$, $B$ is a real symmetric matrix. Let $A^*$ denote the adjoint matrix of $A$, i.e. $AA^*=\det A\cdot I$. I hope the real part of adjoint matrix ${\rm Re}A^*$ is nonnegative. The question is that should I add what kind of condition on $B$ to guarantee ${\rm Re}A^*\geq 0$.

Answer 1

A sufficient condition is as follows. Let $S=Re(Adj(A))$. Choose the diagonal of $B$ positive and large enough. One has

For $n=4p$: $S<0$. For $n=4p+1$, $S>0$.

For $n=4p+2$: $S>0$. For $n=4p+3$, $S<0$.