EDIT 1: I've made a major mistake in the calculations leading up to this question. The question should be regarded as "on hold" until further notice. Sorry for any inconvenience!
EDIT 2: I've updated my question. It it hopefully correct and more transparent now.
I have a real $3 \times 3$ matrix $A$ with entries $a_{ij}.$ I want to find the $9$-dimensional volume of the region satisfying the following three constraints:
- $\mathrm{tr} (A)<0,$
- $\det (A) > b \,\mathrm{tr}(A),$ and
- $\mathrm{tr}(A^TA)\leq 1,$
where $$b=a_{11}a_{22}+a_{11}a_{33}+a_{22}a_{33}-a_{12}a_{21}-a_{13}a_{31}-a_{23}a_{32},$$ and where the third constraint can also be written as $\sum_{i,j=1}^3 a_{ij}^2\leq1,$ i.e., the volume is bounded by the unit $9$-ball.
How should I set up the appropriate integrals for finding this volume? I'm at a bit of a loss here.
A numerical approach would also be of interest, but as I don't know how to set the integrals up in the first place, I don't know how to approach the problem numerically either.
My previous question provides some context.
Thank you.