Solving Linear Equations by Matrix Method (Class 12)

The matrix method solves a system of linear equations by writing it as a single matrix equation AX = B and then multiplying both sides by the inverse of the coefficient matrix: X = A⁻¹B. It works for any number of equations, as long as the number of equations equals the number of unknowns and the determinant of A is not zero.

This method matters because it replaces slow substitution and elimination with one clean formula. Computers use exactly this idea (in refined forms) to solve the thousands of equations that appear in circuit analysis, structural design and economic models.

Key Concepts

• Coefficient matrix A: the n × n matrix of the coefficients of the unknowns, written in order.
• Variable column X: the column matrix of the unknowns, for example [x, y]ᵀ.
• Constant column B: the column matrix of the numbers on the right-hand side.
• Matrix equation: AX = B is a compact way of writing the whole system at once.
• Non-singular condition: the method needs det(A) ≠ 0 so that A⁻¹ exists.

Formulas

• Solution formula: X = A⁻¹B, where A⁻¹ = (1 / det(A)) × adj(A).
• Steps: (1) write the system as AX = B; (2) find det(A) and check it is non-zero; (3) find A⁻¹; (4) multiply X = A⁻¹B.
• Uniqueness: when det(A) ≠ 0, the system has exactly one solution given by this formula.

Worked Example

Solve by the matrix method: x + y = 5 and 2x − y = 4.

1. Write as AX = B with A = [[1, 1], [2, −1]], X = [x, y]ᵀ, B = [5, 4]ᵀ.
2. det(A) = 1 × (−1) − 1 × 2 = −1 − 2 = −3 ≠ 0, so a unique solution exists.
3. The cofactors are C₁₁ = −1, C₁₂ = −2, C₂₁ = −1, C₂₂ = 1, giving adj(A) = [[−1, −1], [−2, 1]].
4. A⁻¹ = (1/−3) × [[−1, −1], [−2, 1]] = [[1/3, 1/3], [2/3, −1/3]].
5. X = A⁻¹B: x = (1/3)(5) + (1/3)(4) = 9/3 = 3; y = (2/3)(5) + (−1/3)(4) = (10 − 4)/3 = 2.
6. Check: 3 + 2 = 5 and 2(3) − 2 = 4. Both equations are satisfied, so (x, y) = (3, 2).

Common Misconceptions

• The method fails when det(A) = 0. With no inverse there is no X = A⁻¹B; the system may have no solution or infinitely many.
• The multiplication order is X = A⁻¹B, not BA⁻¹. Matrix multiplication does not commute, and BA⁻¹ is usually not even defined.
• Variables must be aligned in columns. If x appears under x in every equation, A comes out wrong and so does the answer.

Real-World Applications

• Electrical circuits: solving for currents in networks with many loops using Kirchhoff's laws.
• Structural engineering: finding forces in trusses and frames.
• Economics: balancing supply and demand across several linked markets.
• Computer graphics: solving for transformation parameters that map one set of points to another.

The matrix method turns a system of equations into AX = B and solves it with X = A⁻¹B, provided det(A) ≠ 0. Write the matrices carefully, compute the inverse step by step, and always substitute your answer back into the original equations to confirm.