I have two matrices $A$ with dimension $n\times n$ and $B$ with dimension $n\times m$ that $m<n$.
In my problem matrix, $A$ is fixed and given, but the elements of the $B$ are variables that can be continuous between $0$ and $1$ or discrete binary $0$ or $1$, but it's not important now.
I am working on an equation based on the $A$ and $B$ that is in the blow:
$$x =\int_0^1 e^{At}BB^{T}e^{A^{T}t}{dt}$$
here $t$ is a constant equal to $1$ and my goal is the $\text{Trace}(x^{-1})$.
I want to ask that, is there any property or relation between given matrix $A$ and arbitrary matrix $B$ that by that property or relation I will be able to say that matrix $B1$ will have greater $\text{Trace}(x^{-1})$ than matrix $B2$ ?