The problem I have is I need to optimize $x_k$ and $K$ is given. Let $$Z = \begin{bmatrix} a \sum_{k=1}^{K} k^2 x_k & - b \sum_{k=1}^{K} k x_k \\ - b \sum_{k=1}^{K} k x_k & c \sum_{k=1}^{K} x_k\end{bmatrix}$$.
Now I need to formulate this optimization problem as $trace(Z^{-1})$. Is there a way how the problem could be simplified? All the coefficients are ${a,b,c,d} \in \mathbb{R}^+$ and $x \in \mathbb{R}^+$.
I tried taking the inverse using adjoint and determinant, but it just end up looking more difficult.
It is better to think about such expressions as standard statistical quantities. For example $\mu_0 = \sum_k^K {x_k}$ is just mean of data multiplied by $K$. Define $\mu_n$ as $$\mu_n = \sum_k^K {k^n x_k}$$ Then your matrix is
$$ Z= \begin{bmatrix} a \mu_2 & -b\mu_1 \\ -b\mu_1 & c\mu_0 \end{bmatrix}$$ The inverse is straightforward to calculate.