Redundant and Inconsistent Systems

Determinants and Inconsistant (0 solution) or Redundant (∞ solution) Systems

Given an nxn system of linear equations:

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(i) if  system determinant ∆ ≠ 0 ---> system has a unique solution

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(ii) if system determinant ∆ = 0 and ∆₁ ≠ 0 , ∆₂ ≠ 0 , ...---> system is inconsistent; no solution

ex. 
2 x + 3 y = 6
4 x + 6 y = 10

∆ =

2	3
4	6

∆ = 12 - 12 = 0

∆_x =

6	3
10	0

∆_x = 36 - 30 = 6 ≠ 0

∆_y =

2	6
4	10

∆_y = 20 - 24 = -4 ≠ 0

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(iii) if system determinant ∆ = 0 and at least one of ∆₁ = 0 , ∆₂ = 0 , ... are also zero ---> system is redundant; infinite solutions

ex.
2 x + 3 y = 6
4 x + 6 y = 12

∆ =

2	3
4	6

∆ = 12 - 12 = 0

∆_x =

6	3
12	6

∆_x = 36 - 36 = 0

∆_y =

2	6
4	12

∆_y = 24 - 24 = 0


Note:
ex. what if say ∆ = 5, and ∆_x = 2 and ∆_y = 0
==> No problem!
x = ∆_x / ∆ = 2 / 5 = 0.4
y =  ∆_y / ∆ = 0 / 5 = 0