My question is about the configurations of the planes associated with linear simultaneous questions with 3 variables. When solving a set of equations of the form:
$$ a_1x+b_1y+c_1z=d_1$$ $$ a_2x+b_2y+c_2z=d_2$$ $$ a_3x+b_3y+c_3z=d_3$$
let the matrix A be:
\begin{pmatrix} a_1 & b_1 & c_1 \\\ a_2 & b_2 & c_2 \\\ a_3 & b_3 & c_3 \end{pmatrix}
I am considering the cases when there is no unique solution, i.e. det(A) is zero. In this case, if the geometric configuration is such that two or more of the planes are parallel or the same plane, this configuration can be deduced from A because some rows will be multiples of each other. The matrix will show that some of the planes are at least parallel, and to check if they are the same plane, we then need to check the values of $d_i$. However, det(A) will also be zero if that planes are 1. arranged in a sheaf or 2. if they are the sides of a triangular prism. In these cases though, none of the planes are parallel, so the matrix won't have rows that are multiples of each other. The situations are distinguishable because they are different when solved simultaneously, one is consistent and the other isn't. My question is: Is there a way to deduce from the matrix (in a way similar to the case where some of the planes are parallel, i.e. without solving the equations and eliminating variables), which one of the two configurations the planes are in, given that none of them are parallel or the same plane? Thanks so much!