Matrices and Determinants with Business Applications
Matrices and determinants turn systems of business equations into compact arrays that can be added, multiplied and solved. The exam tests order and conformability, matrix operations, 2 × 2 and 3 × 3 determinants, inverses and Cramer’s rule, usually through sales, cost or resource-allocation problems.
Core rules
- A matrix of order m × n has m rows and n columns. Addition and subtraction require the same order; multiplication AB requires columns of A = rows of B, and the product has order (rows of A) × (columns of B).
- Matrix multiplication is not commutative: AB ≠ BA in general. The identity matrix I leaves a matrix unchanged where the orders allow multiplication.
- For A = [[a, b], [c, d]], det A = ad − bc. If det A ≠ 0, A⁻¹ = (1/det A)[[d, −b], [−c, a]].
- Cramer’s rule for ax + by = e, cx + dy = f uses D = ad − bc, Dx = ed − bf, Dy = af − ec, with x = Dx/D and y = Dy/D when D ≠ 0.
- In business, rows may be products and columns may be branches or months; label units before multiplying so that the result has a sensible meaning.
Worked example
Solve 3x + 2y = 16 and x + 4y = 22. Here D = (3)(4) − (2)(1) = 10. Dx = (16)(4) − (2)(22) = 64 − 44 = 20, so x = 20/10 = 2. Dy = (3)(22) − (16)(1) = 66 − 16 = 50, so y = 50/10 = 5. Check: 3(2) + 2(5) = 16 and 2 + 4(5) = 22. Because D ≠ 0, the system has a unique solution and the coefficient matrix has an inverse.
Traps
- Multiplying matrices in the wrong order or ignoring conformability.
- Using ad + bc for a determinant, or forgetting the sign change in the inverse.
- Assuming a zero determinant means “no solution” without distinguishing inconsistent from dependent systems.