Cramer Rule for 3x3 Matrix

Rule 1

If ∆ ≠ 0. Then the system has unique solution and we can solve the equations by using the formula x = ∆ₓ/∆ , y = ∆ᵧ/∆ ,z = ∆z/∆

Rule 2

If  ∆ = 0 and ∆ₓ= 0 , ∆ᵧ= 0 , ∆z= 0 and at least one of the 2 x 2 minor of ∆ is non-zero,then we can decide the system is consistent and it has infinitely many solution.Now the system of three equations can be reduced into two equations and we can solve by using two suitable equations and assigning an arbitrary values to one of the three unknowns and then solve the other two unknowns

Rule 3

If ∆ = 0 and ∆ₓ= 0 , ∆ᵧ= 0 , ∆z = 0 and all 2 x 2 minor of ∆ is zero but one of the element in ∆ ≠ 0 ,then we can decide the system is consistent and it has infinitely many solution. Now the system of three equations can be reduced into one equation and we can solve by using one suitable equation and assigning an arbitrary values to two unknowns and find the value of remaining one.

Rule 4

If ∆ = 0 and at least one of the values of ∆ₓ, ∆ᵧ ,∆z is not equal to zero. Then we can decide the system is inconsistent and it has no solution.

Rule 5

If ∆ = 0. ∆ₓ = ∆ᵧ= ∆z =0 and all 2 x 2 minors of ∆ is equal to zero and at-least one of 2 x 2 minors of ∆ₓ,∆ᵧ∆z not equal to zero then we can decide the system is inconsistent and it has no solution.

Examples