Let $A$ be an $n \times n$ matrix. Are there formulas that convert linear combinations of traces of powers of $A$ to determinant of $A$ and vice versa from linear combinations of determinants of powers of $A$ to trace of $A$? (I am mainly looking for the latter - conversion of determinants to trace of $A$).



Trace Determinant
You can find the determinant of a matrix $A$ of size $n$ in terms of the traces of $A^m$, for $m = 1, \ldots, n$. The trace of $A^m$ is the sum of the $m$th powers of the eigenvalues of $A$, and you can express the elementary symmetric polynomials (so in particular the product of the eigenvalues, which is the determinant) in terms of the power sums, see for example [here](http://en.wikipedia.org/wiki/Newton%27s_identities).


The converse is not possible. For example, the determinants of all powers of $\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$ and of $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ are 1, but the traces are different.


