\begin{equation} D_1 =\det \begin{pmatrix} \alpha_1 &1& \beta_2\\ \alpha_2 &1& \beta_3\\ \alpha_3 &1& \beta_1 \end{pmatrix} \end{equation} \begin{equation} D_2 = \det \begin{pmatrix} \beta_1 &1& \alpha_2\\ \beta_2 &1& \alpha_3\\ \beta_3 &1& \alpha_1 \end{pmatrix}, \end{equation} \begin{equation} D_3 = \det \begin{pmatrix} \alpha_1 & \alpha_3 & \beta_2\\ \alpha_2 & \alpha_1 & \beta_3\\ \alpha_3 & \alpha_2 & \beta_1 \end{pmatrix}, \end{equation} \begin{equation} D_4 = \det \begin{pmatrix} \alpha_1 & \beta_3 & \beta_2\\ \alpha_2 & \beta_1 & \beta_3\\ \alpha_3 & \beta_2 & \beta_1 \end{pmatrix}, \end{equation}
Note that $\alpha_i, \beta_i$ are real numbers.
Can we relate and connect the four determinants $D_i$, for $i=1,2,3,4$? Can we point out some relations between them?
For example: If $D_1=D_2=0$, then it implies that $D_3=0$ or $D_4=0$.[this property is already known] I am searching for other connections and relationship between D_i.
Something like this but not limited to $D_1 = 0, D_2 \neq 0$, $D_2 \neq 0, D_1 =0$, $D_1 >0, D_2 = 0$ what happens to $D_3, D_4$?
Any interesting properties connecting the four determinants are welcome!
EDIT
Mainly,if we fix $D_1<0$ and $D_2<0$ then what happens to $D_3, D_4$ ? How they behave?
The question has arised from a question on MathOverflow: https://mathoverflow.net/questions/426170/is-it-impossible-for-determinants-of-these-matrices-to-both-be-negative