ECON5005 · Quantitative Tools for Economics
Matrix Algebra
This chapter of the University of Sydney ECON5005 Quantitative Tools for Economics study guide covers matrix algebra, the compact language for the simultaneous relationships that fill economics. You will learn to pack a linear system into a single equation Ax = b, use the determinant to test whether a unique solution exists, and then solve it by the inverse matrix and by Cramer’s rule. It maps to Weeks 10-11 of the Semester 2, 2026 teaching schedule and is examined on the 2-hour final.
What this chapter covers
- 01Compute a 2x2 determinant as ad - bc (down-diagonal minus up-diagonal)
- 02Expand a 3x3 determinant along a row using minors and signed cofactors
- 03Form the 2x2 inverse: swap the diagonal, negate the off-diagonal, divide by det A
- 04Build the general inverse as the adjoint (transposed cofactor matrix) over det A
- 05Read rank as the number of linearly independent rows and link it to invertibility
- 06Use det A = 0 to diagnose no solution vs infinitely many from the right-hand side
- 07Solve Ax = b two ways: x = A-inverse b and Cramer's rule x_i = det(A_i)/det(A)
- 08Set up a national-income macro model in matrix form and solve it by inversion
- 09State det A not equal to 0 as the existence condition for a unique solution
Solve a 2x2 linear system with Cramer's rule
- +1Write the coefficient matrix A = [[5, 3], [2, 4]] and the constants b = [19, 16].
- +1Compute the determinant that goes underneath every ratio: det A = 5(4) - 3(2) = 20 - 6 = 14. It is non-zero, so a unique solution exists.
- +1For x, replace column 1 with b: det A_x = det[[19, 3], [16, 4]] = 19(4) - 3(16) = 76 - 48 = 28, so x = 28/14 = 2.
- +1For y, replace column 2 with b: det A_y = det[[5, 19], [2, 16]] = 5(16) - 19(2) = 80 - 38 = 42, so y = 42/14 = 3.
- +1Check both equations: 5(2) + 3(3) = 10 + 9 = 19 and 2(2) + 4(3) = 4 + 12 = 16 - both hold.
Key terms
- Determinant
- A single number attached to a square matrix. For a 2x2 matrix [[a, b], [c, d]] it equals ad - bc; a non-zero value means the matrix is invertible.
- Minor and cofactor
- The minor M_ij is the smaller determinant left after deleting row i and column j; the cofactor C_ij = (-1)^(i+j) M_ij attaches the chequerboard sign used in expanding a larger determinant.
- Inverse matrix
- The matrix A-inverse satisfying A times A-inverse = I. For a 2x2, A-inverse = (1/(ad - bc)) [[d, -b], [-c, a]], defined only when det A is non-zero.
- Adjoint (adjugate)
- The transpose of the matrix of cofactors. The general inverse is A-inverse = adj(A)/det A, of which the 2x2 swap-and-negate rule is a special case.
- Rank
- The number of linearly independent rows (equivalently columns) of a matrix. A square matrix has full rank exactly when its determinant is non-zero.
- Singular vs non-singular
- A matrix is singular when det A = 0 (no inverse, rank-deficient, dependent rows) and non-singular when det A is non-zero (invertible, full rank, unique solution).
- Cramer's rule
- A formula giving each unknown as a ratio of determinants: x_i = det(A_i)/det(A), where A_i is A with column i replaced by the constant vector b.
- Existence condition
- The requirement det A not equal to 0 that guarantees the system Ax = b has exactly one solution; if det A = 0 the system has either no solution or infinitely many.
Matrix Algebra FAQ
When should I use Cramer's rule and when the inverse matrix?
They give the same answer for a square system, so choose by convenience. Cramer's rule is quick when you only need one or two unknowns, because each is a single ratio of determinants. The inverse matrix is handy when you must solve the same coefficient matrix against several different right-hand sides, since A-inverse is computed once and reused. Either way, compute det A first: if it is zero, neither method applies and the system has no solution or infinitely many.
What does a zero determinant tell me about a system of equations?
A zero determinant means the coefficient matrix is singular: its rows are linearly dependent, its rank is below full, and the system cannot have exactly one solution. Whether it has none or infinitely many is then decided by the right-hand side. If the constants are consistent with the dependency, the equations describe the same line and there are infinitely many solutions; if they are inconsistent, the lines are parallel and there is no solution.
Can AI help me with matrix algebra in ECON5005?
Yes - an AI tutor like Sia is useful for understanding the method. It can walk you step by step through computing a determinant, forming a 2x2 inverse, applying Cramer's rule, and checking a solution back in the original equations, using practice numbers so the moves become automatic. Use it to build understanding and check your reasoning, not to hand in generated answers: ECON5005 is assessed by in-person written exams, so working problems yourself is what pays off, and you should follow the University's academic-integrity rules.
Studying with AI? Sia — free AI economics tutor works through ECON5005 step by step.
Exam move
Make the determinant your reflex first move. On any systems question, compute det A before anything else, because it tells you whether a unique solution exists and whether the matrix can be inverted at all. Then drill the two solution routes until both are automatic: the 2x2 inverse (swap the diagonal, negate the off-diagonal, divide by det A) and Cramer's rule (replace one column with the constants, keep det A underneath). Practise the national-income macro model, since writing it as Ax = b, stating det A = 1 - b not equal to 0, and reading the multiplier 1/(1 - b) is the archetype long question. Always verify by substituting your solution back into the original equations, and write the matrix setup and the existence condition explicitly so you collect method marks even when arithmetic slips. Work on the in-person written format at roughly one minute per mark, and confirm the exact assessment weights and the open/closed-book and calculator rules on your Canvas assessment page.
