It's a bird! It's a plane! It's not superman...

Geometry Level 2

A system of three linear equations with three variables x,y,z x, y, z in the form

{ax+by+cz=dex+fy+gz=hix+jy+kz=l \left\{\begin{matrix} ax+by+cz = d\\ ex+fy+gz = h \\ ix+jy+kz = l \end{matrix}\right.

with (a,b,c,d,e,f,g,h,i,j,k,l)Z(a, b, c, d, e, f, g, h, i, j, k, l) \in \mathbb{Z} could be considered geometrically as three different planes intersecting each other in various ways, depending what those coefficients are. Some of the coefficients may be equal to other coefficients--or not.

Let A \text{A} be the coefficient matrix of that above system. If det(A)=0 \text{det(A)} = 0 , which of the following could be true about the system?

A. The planes are all parallel to each other.

B. A line is formed by the intersection of all three planes.

C. The planes form a triangular prism missing the triangular faces.

D. The planes all intersect at a single point.


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