Suppose we have system of three equations which are non-homogenous . They are basically planes represented in $3$ dimensional coordinate axis. So solution to the system of three linear non homogenous system is equivalent to finding intersection points of planes in the coordinate axis. Now here are the possible outcomes which can happen when three planes intersect :
A) they intersect together at a single point .
B) they intersect together on a common intersection line .
C) all of them intersecting in different common line for each pair .
D) Two planes being parallel and not having any intersectiom point but another plane which cuts both have two common lines with respect to them .
E) All three planes being parallel to each other.
F) all three being same plane.
G) two of them representing same plane while the last one cuts it in common line.
So out of these only
A ) determines a unique point common , which also implies determiant of the coefficient of the variables is non zero and rank of the matrix being 3. While B ) gives a infinite solutions to the system . Which also implies that third system is a linear combination of first two system , and this also tells rank of the coefficient matrix is 2 . Other two D) and E) are cases of system being inconsistent . They all have determinant being zero but also that rank is 2 and 1 respectively.
While F) and G) are again consistent system which have infinite solutions and det being zero and rank respectively 1 and 2 . I most probably understood and gave correct description to all these cases except the case of C where i am not getting what can be said about determinant and also rank as such all three would look like its all different equations so deteminant is non zero but then if i apply cramer rule here then it will have a solution which contradicts the fact shown by graph thats its inconsistent . C case graph , this shows the three planes intersection top view which are being formed :
Best Answer
If you intersect the planes two at a time, you’ll get three lines, in general. Case (C) describes a configuration where these three lines are parallel and distinct, so there is no point where all three planes intersect.
Geometrically, the normal vectors of the three planes are coplanar (i.e. linearly dependent), and this means that the determinant formed from these three vectors is zero.