Coefficients of characteristic polynomial of a $3\times 3$ matrix Let $A$ be a $3\times 3$ matrix over reals. Then its characteristic polynomial $\det(xI-A)$ is of the form $x^3+a_2x^2+a_1x+a_0$. It is well known that 
$$-a_2=\mbox{trace}(A) \mbox{ and } -a_0=\det(A).$$
Note that these constants are expressed as functions of $A$ without referring to eigenvalues of $A$. 
Q. What is interpretation of $a_1$ in terms of $A$ without considering its eigenvalues?

This could be trivial, but I have never seen it.

For invertible $A$, we have Adj$(A)=det(A)\cdot A^{-1}$ so the trace of the Adj$(A)$ is $det(A)\cdot (\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3})=x_1\cdot x_2+x_1\cdot x_3+x_2\cdot x_3=a_1 $ where $x_i$ are the eigenvalues. Since it holds for invertible $A$, it holds for all matrices.
