1st Year Math Chapter 4 Exercise 4.3 Notes: Solving Systems of Linear Equations Using Matrices (Punjab Board 2026-27)
Complete concept notes, formulas, and solved examples for ICS/FSc Part 1 Mathematics, Punjab Board (PECTAA), Single National Curriculum 2026-27 session.
Exercise 4.3 closes the Matrices and Determinants chapter by applying the determinant and inverse concepts from Exercise 4.2 to solve systems of linear equations, using both the matrix inversion method and Cramer’s rule.
What Does This Exercise Cover?
This exercise shows how a system of two linear equations can be written as a single matrix equation, and how that equation can be solved either by using the inverse matrix or by using ratios of determinants (Cramer’s rule).
Key Concepts
1. Writing a System as a Matrix Equation
A system of linear equations can be written in the form AX = B, where A is the matrix of coefficients, X is the column matrix of unknowns, and B is the column matrix of constants.
2. The Matrix Inversion Method
If A is non-singular, the system AX = B can be solved directly as X = A⁻¹B, using the inverse of A found in Exercise 4.2.
3. Cramer’s Rule
Cramer’s rule solves for each variable using a ratio of determinants: replace the column of coefficients for that variable in A with the constants from B to form a new determinant, then divide by det(A).
4. When a Unique Solution Exists
Both methods require det(A) ≠ 0. If det(A) = 0, the system does not have a unique solution — it may have no solution or infinitely many, depending on the equations.
Cramer’s Rule for Two Variables
For the system a₁x + b₁y = c₁ and a₂x + b₂y = c₂:
| Determinant | Formed From |
| D | |a₁ b₁; a₂ b₂| (the coefficient matrix) |
| Dx | |c₁ b₁; c₂ b₂| (x-column replaced with constants) |
| Dy | |a₁ c₁; a₂ c₂| (y-column replaced with constants) |
Solved Examples
Example 1: Solve 2x + y = 5 and 3x + 2y = 8 using the matrix inversion method.
Write A = [[2, 1], [3, 2]], X = [[x], [y]], B = [[5], [8]].
det(A) = (2×2) − (1×3) = 4 − 3 = 1.
adj(A) = [[2, −1], [−3, 2]], so A⁻¹ = [[2, −1], [−3, 2]].
X = A⁻¹B = [[2×5 + (−1)×8], [−3×5 + 2×8]] = [[10−8], [−15+16]].
Answer: x = 2, y = 1
Example 2: Solve x + 2y = 4 and 3x − y = 5 using Cramer’s rule.
D = |1 2; 3 −1| = (1×−1) − (2×3) = −1 − 6 = −7.
Dx = |4 2; 5 −1| = (4×−1) − (2×5) = −4 − 10 = −14.
Dy = |1 4; 3 5| = (1×5) − (4×3) = 5 − 12 = −7.
x = Dx/D = −14/−7 = 2, y = Dy/D = −7/−7 = 1.
Answer: x = 2, y = 1
Sample MCQs
1. A system of linear equations can be written in matrix form as:
a) A + X = B b) AX = B c) A − X = B d) AB = X
Answer: b) AX = B
2. In the matrix inversion method, the solution is given by:
a) X = AB b) X = A⁻¹B c) X = BA⁻¹ d) X = B⁻¹A
Answer: b) X = A⁻¹B
3. Cramer’s rule requires computing:
a) Only one determinant b) The determinant of A, plus one determinant per variable c) The inverse of B d) The adjoint of X
Answer: b) The determinant of A, plus one determinant per variable
4. If det(A) = 0 in a system AX = B, the matrix inversion method:
a) Always works b) Cannot be used, since A⁻¹ doesn’t exist c) Gives x = 0 automatically d) Gives infinite solutions automatically
Answer: b) Cannot be used, since A⁻¹ doesn’t exist
5. In Cramer’s rule, the value of x is found as:
a) D / Dx b) Dx / D c) Dx × D d) D − Dx
Answer: b) Dx / D
Important Short Questions
- Write the system 3x + y = 7, x − 2y = 1 in matrix form AX = B.
- State the formula for solving a system using the matrix inversion method.
- Explain how Cramer’s rule uses determinants to find each variable.
- What condition must be satisfied for the matrix inversion method to work?
- Using Cramer’s rule, find only the value of x for: 2x + 3y = 8, x − y = 1.
Important Long Questions
- Solve the system 2x + y = 5, 3x + 2y = 8 using the matrix inversion method, showing all steps.
- Solve the system x + 2y = 4, 3x − y = 5 using Cramer’s rule, showing all steps.
- Explain what it means for a system of linear equations to have no unique solution, in terms of the determinant of A.
- Compare the matrix inversion method and Cramer’s rule as approaches to solving a system of two linear equations.
How to Approach This Exercise Effectively
- Set up A, X, and B carefully before choosing a method — a mismatched row or sign here will carry through the entire problem.
- For the matrix inversion method, reuse the same adjoint/inverse process from Exercise 4.2 rather than treating it as a new skill.
- In Cramer’s rule, replace only one column at a time (x-column for Dx, y-column for Dy) — mixing this up is the most common mistake.
- Always check det(A) first — if it’s zero, state clearly that no unique solution exists rather than attempting either method.
- Verify your final x and y values by substituting them back into both original equations.
FAQs
Q: What is the difference between the matrix inversion method and Cramer’s rule?
A: The matrix inversion method solves for both variables at once using X = A⁻¹B, while Cramer’s rule solves for each variable individually using a separate ratio of determinants.
Q: What happens if det(A) = 0 when solving a system?
A: Neither method can produce a unique solution — the system either has no solution or infinitely many, depending on the specific equations.
Q: Can these methods be extended to three variables?
A: Yes — both the matrix inversion method and Cramer’s rule extend naturally to larger systems, using 3×3 (or larger) determinants and matrices.
Q: Which method is faster for a two-variable system?
A: Cramer’s rule is often quicker for just two variables, since it avoids computing a full inverse matrix — but both methods give identical results.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 4: Matrices and Determinants, Exercise 4.3.