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MathsMediumClass 12

System of Linear Equations — Cramer's Rule

Matrices & Determinants

44

JEE Qs

10%

Hard

60

min

Master determinant calculations and the precise conditions for unique, infinite, and no solutions to apply Cramer's Rule effectively in problem-solving.

🧮 Key Formulas

For a system a₁x + b₁y + c₁z = d₁, a₂x + b₂y + c₂z = d₂, a₃x + b₃y + c₃z = d₃:
D = | a₁ b₁ c₁ | = a₁(b₂c₃ - b₃c₂) - b₁(a₂c₃ - a₃c₂) + c₁(a₂b₃ - a₃b₂)
| a₂ b₂ c₂ |
| a₃ b₃ c₃ |
Dₓ = | d₁ b₁ c₁ | (d₁ replaces a₁ column in D)
| d₂ b₂ c₂ |
| d₃ b₃ c₃ |
Dᵧ = | a₁ d₁ c₁ | (d₁ replaces b₁ column in D)
| a₂ d₂ c₂ |
| a₃ d₃ c₃ |
D₂ = | a₁ b₁ d₁ | (d₁ replaces c₁ column in D)
| a₂ b₂ d₂ |
| a₃ b₃ d₃ |
If D ≠ 0: Unique solution x = Dₓ/D, y = Dᵧ/D, z = D₂/D.
If D = 0:
If Dₓ = 0 AND Dᵧ = 0 AND D₂ = 0: Infinite solutions.
If at least one of Dₓ, Dᵧ, D₂ is non-zero: No solution.
For a homogeneous system (d₁ = d₂ = d₃ = 0):
If D ≠ 0: Only trivial solution (x=0, y=0, z=0).
If D = 0: Non-trivial solutions exist (infinite solutions).

✅ Key Points for JEE

  • 1Cramer's Rule is primarily for systems where the number of linear equations equals the number of variables (square systems).
  • 2The determinant D of the coefficient matrix is crucial; its value dictates the primary nature of the solution (unique vs. non-unique).
  • 3When D=0, it is mandatory to evaluate Dₓ, Dᵧ, D₂ to differentiate between infinite solutions (all are zero) and no solution (at least one is non-zero).
  • 4For homogeneous systems, D=0 is the condition for non-trivial solutions, which are always infinite if they exist beyond the trivial one.
  • 5Systematic and accurate calculation of determinants is paramount, especially for 3x3 matrices, as sign errors are common.

⚠️ Common Mistakes

  • ✕Sign errors or arithmetic mistakes during the expansion and evaluation of 3x3 determinants D, Dₓ, Dᵧ, D₂.
  • ✕Incorrectly forming Dₓ, Dᵧ, or D₂ by replacing the wrong column with the constant terms.
  • ✕Confusing the conditions for infinite solutions (D=0 AND Dₓ=Dᵧ=D₂=0) with no solution (D=0 AND at least one of Dₓ, Dᵧ, D₂ is non-zero).
  • ✕Forgetting to check Dₓ, Dᵧ, D₂ when D turns out to be zero, leading to an incomplete or incorrect conclusion about the system's consistency.

📝 Practice Questions

See all

Q15. x + y + 2z = 6 If the system of linear equations : 2x + 3y + az = a + 1 where a, b ∈R, has infinitely many solutions, then −x −3y + bz = 2 b 7a + 3b is equal to : (1) 16 (2) 12 (3) 22 (4) 9 = 0, y ∈(−π2 , π2 ) with

2025·MCQMedium

Q5. For some n ≠10, let the coefficients of the 5 th, 6 th and 7 th terms in the binomial expansion of (1 + x)n+4 be in A.P. Then the largest coefficient in the expansion of (1 + x)n+4 is: (1) 20 (2) 10 (3) 35 (4) 70

2025·MCQMedium

Q21.Let A be a square matrix of order 3 such that det(A) = −2 and det(3 adj(−6 adj(3A))) = 2m+n ⋅3mn, m > n . Then 4 m + 2n is equal to _______ , then m −n is equal to _______

2025·NumericalHard

Q1. For a 3 × 3 matrix M , let trace (M) denote the sum of all the diagonal elements of M . Let A be a 3 × 3 matrix such that |A| = 12 and trace (A) = 3. If B = adj(adj(2A)), then the value of |B|+ trace (B) equals : (1) 56 (2) 132 (3) 174 (4) 280

2025·MCQMedium

Q10. x + y + z = 6 The system of equations x + 2y + 5z = 9, has no solution if x + 5y + λz = μ, (1) λ = 15, μ ≠17 (2) λ ≠17, μ ≠18 (3) λ = 17, μ ≠18 (4) λ = 17, μ = 18

2025·MCQMedium

Q2. x + 2y −3z = 2 If the system of equations 2x + λy + 5z = 5 has infinitely many solutions, then λ + μ is equal to : 14x + 3y + μz = 33 (1) 13 (2) 10 (3) 12 (4) 11 and

2025·MCQMedium

NCERT Chapters

  • Class 12 Mathematics Ch 4: Determinants