I would like to calculate the determinant of some $3\times3$ matrix $$\boldsymbol{A}(t) = \int_{0}^{t}\boldsymbol{B}(s-t)\boldsymbol{C}(s-t)\boldsymbol{D}(s-t)ds$$ in terms of the determinants of the $3\times3$ matrices $\boldsymbol{B}(z), \boldsymbol{C}(z), \boldsymbol{D}(z)$. In addition $\boldsymbol{A}$ is a symmetric and positive definite matrix. Ideally, there would be a relation to calculate $\det(A)$ exactly in terms of determinants of $\boldsymbol{B}(z), \boldsymbol{C}(z), \boldsymbol{D}(z)$, is this possible?
Otherwise, it would be important to also know if $\det(\boldsymbol{A}(t)) >0$ at some point $t$. Could the Jensen inequality be used to firstly state that $$\det(\boldsymbol{A}(t)) \geq \int_{0}^{t}\det(\boldsymbol{B}(s-t))\det(\boldsymbol{C}(s-t))\det(\boldsymbol{D}(s-t)) dt$$ (and if the RHS is positive then this is enough)? Or would there be another way to determine if $det(\boldsymbol{A}(t)) >0$ from the determinants of $\boldsymbol{B}(z), \boldsymbol{C}(z), \boldsymbol{D}(z)$?
Many thanks