Let there be the ring $A = \mathbb{C}[x_{ij}, y_{ij}\mid i,j \in {1,2}] $. It's coordinate ring of 8 variables.
Consider $I$ generated by $4$ elements $x_{11}y_{11}+x_{12}y_{21}$, $x_{11}y_{12} + x_{12}y_{22}$, $x_{21}y_{11}+x_{22}y_{12}$, $x_{21}y_{12}+x_{22}y_{21}$.
In other words, $I$ is generated by entries of the matrix product $$\begin{pmatrix}x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix} \begin{pmatrix}y_{11} & y_{12} \\ y_{21} & y_{22} \end{pmatrix}.$$
I want to find the equations that describe the irreducible components of vanishing set $V(I)$, where $V(I)$ is the set of $\mathbb{A_\mathbb{C}^8}$ where all the polynomials in $I$ vanish. Irreducible components of $V(I)$ are the components coinciding to minimal prime ideals.
Observation: after substitution of $x,y$ by an array from vanishing set, we obtain a zero product matrix, that means, that either all $x$'s or all $y$'s equal 0, or both matrices have rank $1$. So, first matrix leaves a non-zero eigenvector that is going to zero after second matrix multiplication.
I get $ay_{11}+by_{12}=ay_{21}+by_{22}$ and $ax_{11}+bx_{12}=ax_{12}+bx_{22}=0$ for some $(a,b)$.
However, I don't know how to go about it.