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MathidiaMathematics Academy
Matrices and transformations
Lesson 8 of 75
Standard practice

Linear systems

Objective. Solve simultaneous equations with matrices.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

A system of linear equations can be written as a matrix equation Mx = v, where M contains the coefficients, x the unknowns, and v the constants. If M is invertible, the unique solution is x = M⁻¹v, found by multiplying both sides on the left by M⁻¹. If det(M) = 0, the system either has no solutions or infinitely many, depending on whether the equations are consistent. This connects directly to matrix inverses: solving simultaneous equations becomes a single matrix calculation rather than elimination by hand.

Worked example

Use matrices to solve the linear system 2x + y = 5, 3x + 2y = 8.

2 … 5
Coordinate grid with the line 2 … 5
  1. 1Write as M = [[2,1],[3,2]], v = [5,8], so Mx = v
  2. 2det(M) = (2×2) − (1×3) = 4 − 3 = 1, so M⁻¹ = [[2,−1],[−3,2]]
  3. 3x = M⁻¹v gives x-value = (2×5) + (−1×8) = 10 − 8 = 2
  4. 4y-value = (−3×5) + (2×8) = −15 + 16 = 1

Answer: x = 2, y = 1

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Common mistake. Students multiply Mx = v by M⁻¹ on the wrong side, writing x = vM⁻¹. Matrix multiplication order matters, so M⁻¹ must multiply on the left of both sides consistently: x = M⁻¹v.

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