Solving equations with matrices / Cramer's rule: Class 12 practice
A system AX = B with |A| ≠ 0 has the unique solution X = A⁻¹B. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- 2×2 determinants
The idea behind solving equations with matrices / cramer's rule
A system AX = B with |A| ≠ 0 has the unique solution X = A⁻¹B. For two equations, Cramer's rule gives x = D₁/D and y = D₂/D.
Step-by-step method
- Write the coefficient determinant D.
- Replace columns by constants to get D₁, D₂.
- x = D₁/D, y = D₂/D; check in both equations.
Worked example 1 (Basic)
Solve 3x − 2y = -11 and -4x + 4y = 12 using determinants.
- D = 4
- D₁ = -20, D₂ = -8
- x = -5, y = -2
- Answer: x = -5, y = -2
Worked example 2 (Standard)
Solve 4x − 5y = -10 and -5x + 3y = -7 using determinants.
- D = -13
- D₁ = -65, D₂ = -78
- x = 5, y = 6
- Answer: x = 5, y = 6
Worked example 3 (Challenge)
Solve 4x + 2y = 16 and 4x + 3y = 20 using determinants.
- D = 4
- D₁ = 8, D₂ = 16
- x = 2, y = 4
- Answer: x = 2, y = 4
Common mistakes and misconceptions
- Replacing the wrong column
- Dividing by the wrong determinant
Practice questions
- (Basic) Solve 4x + 0y = 12 and 1x + 2y = -1 using determinants.
- (Basic) Solve 6x + 0y = 0 and 3x + 5y = 15 using determinants.
- (Standard) Solve 6x + 0y = -12 and -1x + 1y = 8 using determinants.
- (Standard) Solve 5x + 1y = -9 and 1x + 2y = 0 using determinants.
- (Challenge) Solve 4x − 5y = 1 and -4x + 3y = 9 using determinants.
- (Challenge) Solve 4x − 3y = 9 and -2x + 2y = -6 using determinants.
Answer key with full working
- Q1: D = 8 → D₁ = 24, D₂ = -16 → x = 3, y = -2 → Answer: x = 3, y = -2
- Q2: D = 30 → D₁ = 0, D₂ = 90 → x = 0, y = 3 → Answer: x = 0, y = 3
- Q3: D = 6 → D₁ = -12, D₂ = 36 → x = -2, y = 6 → Answer: x = -2, y = 6
- Q4: D = 9 → D₁ = -18, D₂ = 9 → x = -2, y = 1 → Answer: x = -2, y = 1
- Q5: D = -8 → D₁ = 48, D₂ = 40 → x = -6, y = -5 → Answer: x = -6, y = -5
- Q6: D = 2 → D₁ = 0, D₂ = -6 → x = 0, y = -3 → Answer: x = 0, y = -3
Syllabus note
This skill is listed under "Determinants" for Class 12 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is solving equations with matrices / cramer's rule in?
"Determinants", Class 12 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.