ECON1003 · Quantitative Methods In Economics
Linear Algebra
A matrix is a grid of numbers, sized rows × columns, and linear algebra is how ECON1003 solves systems of equations compactly. Addition and scalar multiplication are element-wise and need matching sizes; matrix multiplication is the odd one out — row-of-A dotted into column-of-B, defined only when A's columns equal B's rows, and not commutative (AB ≠ BA). On the cumulative final you face a multiplication or dimension check, a system solved by Gaussian / Gauss–Jordan elimination, a determinant (2×2, and the 3×3 with the + − + sign pattern), Cramer's rule, and an inverse (by elimination or cofactors) used to solve X = A⁻¹B. The matrix methods are not on the formula sheet — you memorise them — so this chapter is pure technique you drill to automatic.
What this chapter covers
- 017.1 Dimension, transpose and special matrices
- 027.2 Matrix arithmetic and multiplication (AB ≠ BA)
- 037.3 Gaussian and Gauss–Jordan elimination
- 047.4 Determinants: 2×2 and the 3×3 sign pattern
- 057.5 Cramer's rule
- 067.6 The inverse and solving X = A⁻¹B
Worked example: solve a 2×2 system by Cramer's rule
- +1Write the coefficient matrix and find its determinant. A = [[2, 3], [4, −1]]; det A = (2)(−1) − (3)(4) = −2 − 12 = −14.
- +1Form Aₓ by replacing the x-column with the constants. Aₓ = [[13, 3], [5, −1]]; det Aₓ = (13)(−1) − (3)(5) = −13 − 15 = −28.
- +1Form Aₑ by replacing the y-column with the constants. Aₑ = [[2, 13], [4, 5]]; det Aₑ = (2)(5) − (13)(4) = 10 − 52 = −42.
- +1Apply Cramer's rule. x = det Aₓ/det A = −28/−14 = 2; y = det Aₑ/det A = −42/−14 = 3.
- +1Check by substitution. 2(2) + 3(3) = 4 + 9 = 13 ✓ and 4(2) − 3 = 5 ✓.
Key terms
- Matrix multiplication
- Row-into-column: (AB)ᵢⲤ = Σᵣ aᵢᵣbᵣⲤ. Defined only when A's columns equal B's rows; an m×n times an n×k gives an m×k. It is not commutative: AB ≠ BA in general.
- Determinant
- A single number from a square matrix that tells you whether it is invertible (det ≠ 0). The 2×2 is ad − bc; the 3×3 expands along a row using the + − + sign pattern of cofactors.
- Gauss–Jordan elimination
- Row operations that reduce an augmented matrix to reduced row-echelon form, reading the solution straight off. Gaussian elimination stops at upper-triangular form and back-substitutes.
- Cramer's rule
- A formula for solving a square system: each variable equals the determinant of A with its column replaced by the constants, divided by det A. It works only when det A ≠ 0.
- Inverse matrix (A⁻¹)
- The matrix with A⁻¹A = I. Found by elimination or by cofactors (A⁻¹ = Cᵀ/det A). It solves a system in one move: X = A⁻¹B, provided det A ≠ 0.
Linear Algebra FAQ
Why is AB not the same as BA?
Matrix multiplication is row-of-A dotted into column-of-B, so the order changes which numbers meet. Even for two 2×2 matrices, AB and BA generally give different results — the course shows A = [[1,2],[3,4]], B = [[5,6],[7,8]] where AB = [[19,22],[43,50]] but BA = [[23,34],[31,46]]. Always do the dimension check first: A's columns must equal B's rows.
When can I use Cramer's rule or the inverse?
Only when the coefficient matrix is square and its determinant is non-zero (det A ≠ 0). A zero determinant means the matrix is singular — no unique solution — so Cramer's rule and the inverse both fail, and you fall back on elimination to see whether the system has no solution or infinitely many.
Are the matrix methods on the formula sheet?
No. Unlike the calculus and financial-maths formulas, the matrix methods — multiplication, elimination, the determinant expansions, Cramer's rule and the inverse — are NOT on the provided sheet. You must memorise the procedures, especially the 3×3 determinant's + − + sign pattern, which is a frequent slip.
Exam move
Linear algebra is pure, memorisable technique with nothing on the formula sheet, so it is the most 'bankable' topic if you drill it. Always run the dimension check before multiplying, and keep AB ≠ BA front of mind. For systems, know two routes — elimination (Gaussian/Gauss–Jordan) and the determinant family (Cramer's rule, the inverse) — and remember the determinant family needs det A ≠ 0. The 3×3 determinant's + − + sign pattern is the single most common slip, so practise it until it is automatic. Check every solution by substituting back; it is fast and catches arithmetic errors before they cost marks.
