TITLE: How to get determinant of $A$ in terms of tr$(A^k)$?
QUESTION [5 upvotes]: Suppose that $A$ is $n$-square  matrix such that $t_r:=$ tr$(A^r), r=1, 2, \cdots, n$ are given real numbers. How shall we  compute $\det(A)$ in terms of $t_r$s?
I am completely unable to do this. Please help me. Thanks in advance

REPLY [2 votes]: Something easy to implement with a computer algebra system: 
$\det(A)$ is the coefficient of $t^n$ in the Taylor series of 
$$ \exp(\   \mathrm{tr}(A)t -  \frac{\mathrm{tr}(A^2)}{2} t^2  + \ldots + (-1)^{n-1} \frac{\mathrm{tr}(A^n)}{n} t^n\ )$$
at $t=0$.