What exactly is the relation between inconsistency, determinant being non-zero and number of solutions in a system of three equations in three variables matrix?
I mean these are confusing statements given in my textbook :
1)det A = 0, implies the system has infinitely many solutions (consistent) or no solution(inconsistent)
2)det A not equal to zero implies solution is unique and consistent.
Thus far is okay.
Now, (1) when det not equal to zero, three planes intersect at exactly one point(consistent)
2)when det not equal to zero : either infinite solutions(consistent) means at least one plane is dependent on the others(parallel, overlapping or a combination????)
i.e three planes might intersect in a line or all three are identical (same plane) or two planes coincide and the third intersect them(intersecting line gives infinite solutions.
OR
no solution(inconsistent) :
three planes might be parallel,
pairwise intersection without meeting at a common point(like a triangular prism)
two planes intersect in a line, but the third plane is parallel to that line
In all these no single point satisfies all three equations.
What I don't understand : Can two planes coincide and third intersect in a point? In that case is the det non-zero? and solution consistent?
I mean these are confusing statements given in my textbook :
1)det A = 0, implies the system has infinitely many solutions (consistent) or no solution(inconsistent)
2)det A not equal to zero implies solution is unique and consistent.
Thus far is okay.
Now, (1) when det not equal to zero, three planes intersect at exactly one point(consistent)
2)when det not equal to zero : either infinite solutions(consistent) means at least one plane is dependent on the others(parallel, overlapping or a combination????)
i.e three planes might intersect in a line or all three are identical (same plane) or two planes coincide and the third intersect them(intersecting line gives infinite solutions.
OR
no solution(inconsistent) :
three planes might be parallel,
pairwise intersection without meeting at a common point(like a triangular prism)
two planes intersect in a line, but the third plane is parallel to that line
In all these no single point satisfies all three equations.
What I don't understand : Can two planes coincide and third intersect in a point? In that case is the det non-zero? and solution consistent?