Problem statement: Find the singular locus of $V(J) \subseteq \mathbb{A}^{7}$ where $J$ is the ideal
$$(A_1 D_1 - S A_2 D_2, B_1 D_1 -S B_2 D_2, A_1 D_2 + A_2 D_1, B_1 D_2 + B_2 D_1).$$
Find the irreducible components.
The singular locus is defined as the set of singular points of $V$. Usually I am given a hypersurface to calculate the singular points but in this case I have an ideal. Do I then need to solve a system of partial derivatives (equal to 0) for each polynomial generating the ideal?
Note: I am allowed to use Macaulay2, but I am not good at it at all.
Edit: I followed the hint(s) given in the comments by Tabes Bridges. The first step is to find the Jacobian matrix $(\frac{\partial f_j}{\partial x_j})$. Thus, let
$$f_1 = A_1 D_1 - SA_2 D_2 \quad \quad \quad \quad f_2 = B_1 D_1 - S B_2 D_2,$$
$$f_3 = A_1 D_2 + A_2 D_1 \quad \quad \quad \quad f_4 = B_1 D_2 + B_2 D_1.$$
Note that since I don't know how the coordinates are given (i.e. in which order s.t. $f_i \in k[X_1, X_2, \ldots, X_7]$) I chose $X_1 = A_1, X_2 = A_2, X_3 = B_1, X_4 = B_2, X_5 = D_1, X_6 = D_2, X_7 = S$. Then the Jacobian matrix becomes
$$\begin{pmatrix} D_1 & -S D_2 & 0 & 0 & A_1 & -S A_2 & -A_2 D_2 \\ 0 & 0 & D_1 & -S D_2 & B_1 & -S B_2 & -B_2 D_2 \\ D_2 & D_1 & 0 & 0 & A_2 & A_1 & 0 \\ 0 & 0 & D_2 & D_1 & B_2 & B_1 & 0 \end{pmatrix}.$$
The next step would be to find the vanishing minors. If I recall correctly, then minors are the determinants of submatrices. The issue here is that there seem to be far too many submatrices to consider doing by hand.