Cramer's rule

If Δ =  ≠ 0, then the solution of the system of non-homogeneous simultaneous linear equations

a₁x + b₁y + c₁z = Δ₁

a₂x + b₂y + c₂z = Δ₂

a₃x + b₃y + c₃z = Δ₃

is shown by {where (Δ₁, Δ₂, Δ₃) ≠ (0, 0, 0)}

If any  of  Δ_x, Δ_y, Δ_z ∈ R  and Δ ≠ 0, system of equation will have general solution and is called consistence independent

If   Δ_x = Δ_y = Δ_z= 0 and  Δ is also zero then the system of equations will have infinitely several solutions and is called consistence dependent.

If Δ_x, Δ_y, D_z is non zero and Δ is zero, then the system of equations will have no  solution and  is called inconsistant.

Example:   For what values of p and q, the system of equations

                        2x + py + 6z = 8           

                        x + 2y + qz = 5

                        x + y + 3z = 4

                        has (i) no solution (ii) a unique solution (iii) infinitely many solutions

Solution:       Δ =  = 2(6 - q) - p(3 - q) + 6(1 - 2)

                        = 12 - 2q - 3p + pq  - 6 = pq - 2q - 3p + 6 = (p -2)(q -3)

                        Δ₁ =  = 8(6 -q) - p(15 - 49) + 6(5 - 8)

                        = 48 - 8q - 15p + 4pq -18 = 4pq - 8q - 15p + 30

                        = 4q(p - 2) - 15(p -2) = (4q - 15)(p -2)

                        Δ₂ =  = 2(15 - 4q) - 8(3 - q) + 6(4 - 5) = 0

                        Δ₃ =  = 2(8 -5) - p(4 - 5) + 8(1 - 2)  = p -2

                        Case -I: when q = 3,  p  2, Δ_z = 0, Δ₁ ≠ 0.

                         provided system of equations will have no solution.

                        Case-II: When Δ ¹ 0 , i.e.  p ≠ 2, q ≠ 3, 

                        ∴ provided system of equation has unique solution

                        Case-III: When Δ = 0, i.e.  p = 2, or q = 3

                        When p = 2,  Δ₁ = 0, Δ₂ = 0, Δ₃ = 0

                  ∴ provided system of equation has infinitely many solutions.

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