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Symmetric polynomials of the roots of a polynomial can be written aspolynomials of the coefficients, and by applying this to the characteristicpolynomial we can write a symmetric polynomial of the eigenvalues $a_{i}$ of an$n\times n$ matrix $A$ as a polynomial of the entries of the matrix. We give amagic formula for this: symbolically substitute $a\mapsto A$ in the symmetricpolynomial and replace multiplication by $\det$. For instance, for a $2\times2$matrix $A$ with eigenvalues $a_{1},a_{2}$, \begin{align*} a_1 a_2^2 +a_1^2 a_2& =\det(A_1, A_2^2)+ \det(A_1^2, A_2) \end{align*} where $A_i^k$ is the $i$-thcolumn of $A^k$. One may also take negative powers, allowing us to calculate:\begin{align*} a_1a_2^{-1}+a_1^{-1}a_{2} &=\det(A_{1},A_{2}^{-1})+\det(A_1^{-1},A_{2}) \end{align*} The magic method alsoworks for multivariate symmetric polynomials of the eigenvalues of a set ofcommuting matrices, e.g. for $2\times2$ matrices $A$ and $B$ with eigenvalues$a_1,a_2$ and $b_{1},b_{2}$, \begin{align*} a_1 b_1 a_2^2 + a_1^2a_2b_2 & =\det(AB_{1},A_2^2) + \det(A_1^2,AB_2) \end{align*}

A magic determinant formula for symmetric polynomials of eigenvalues