Let $M$ be a symmetric $n \times n$ matrix.

Is there any equality or inequality that relates the trace and determinant of $M$?

**Answer**

Not exactly what you’re looking for but I would be remiss not to mention that for any complex square matrix $A$ the following identity holds:

$$\det(e^A)=e^{\mbox{tr}(A)} $$

**Attribution***Source : Link , Question Author : TPArrow , Answer Author : Rodrigo de Azevedo*