Chapter 9 · Test 2
Determinants & Inverses for Economics
Determinants, inverse matrices, nonsingularity, and Cramer's rule.
What you will learn
- Calculate a 2 × 2 determinant
- Compute an inverse-matrix entry
- Solve a nonsingular system
Worked example
Calculate a 2 × 2 determinant: follow the method step by step
Try the problem first, then compare your approach with the complete solution and the reasoning behind it.
Calculate a 2 × 2 determinant
Find det(A) for A = [[3, 2], [1, 4]].
For [[a, b], [c, d]], det(A) = ad − bc
In plain English: For a two-by-two matrix, the determinant is the product along the main diagonal minus the product along the other diagonal.
Bold blue terms in this symbol guide include definitions. Hover, focus, or tap a term. Each row also shows how to read the notation aloud.
- A
- the 2 × 2 matrixHow to read: A
- det(A)
- the Determinant: A number calculated from a square matrix that indicates, among other things, whether the matrix is invertible. of AHow to read: the determinant of A
- a, b, c, d
- the four entries of A = [[a,b],[c,d]]How to read: a, b, c, and d
Solution
- Identify a = 3, b = 2, c = 1, and d = 4.
- Compute ad = 3(4) = 12.
- Compute bc = 2(1) = 2.
- Subtract: det(A) = 12 − 2 = 10.
Logic and solving tips
- A nonzero determinant means the matrix has an inverse and its associated square system has a unique solution.
- Keep the order ad − bc; reversing the two diagonal products changes the sign.
Where economists use it
A determinant tells an economist whether a simultaneous demand-and-supply system has one recoverable equilibrium or cannot be uniquely solved from the stated equations.
Key idea: A nonzero determinant means the matrix has an inverse and its associated square system has a unique solution.
Applied case study
Testing a two-sector economy for invertibility
Scenario
A 2-by-2 determinant quickly shows whether a small simultaneous economic system is invertible. Test the matrix from the simplified agriculture-manufacturing input-output model.
Problem
Find det(B) for B = [[0.8, -0.2], [-0.1, 0.6]] and state what it implies.
det(B) = ad - bc
In plain English: The determinant summarizes whether the matrix transformation is invertible; a zero determinant means the associated square system does not have a unique solution.
Worked solution
- Compute the main-diagonal product: 0.8(0.6) = 0.48.
- Compute the other diagonal product: (-0.2)(-0.1) = 0.02.
- Subtract: det(B) = 0.48 - 0.02 = 0.46.
- Because the determinant is nonzero, B is invertible.
Economic interpretation: The teaching economy's sector-balance equations have one algebraic output solution for a given final-demand vector.
Source: U.S. Bureau of Economic Analysis, Concepts and Methods of the U.S. Input-Output Accounts BEA supplies the real input-output setting. The 2-by-2 matrix is a classroom abstraction, not a published BEA coefficient matrix.
Reasoning habits that help
- A nonzero determinant means the matrix has an inverse and its associated square system has a unique solution.
- For a 2 × 2 inverse, the original top-left entry becomes the inverse's bottom-right entry after division by the determinant.
- Cramer's rule and elimination produce the same unique solution when the coefficient determinant is nonzero.
Where economists use these methods
- A determinant tells an economist whether a simultaneous demand-and-supply system has one recoverable equilibrium or cannot be uniquely solved from the stated equations.
- An input-output economist uses a matrix inverse to calculate both the direct and upstream production required to satisfy a change in final demand.
- An economist writes equilibrium price or output as a ratio of determinants to see directly how model coefficients and policy parameters affect the solution.
Chapter vocabulary
Study 21 key terms
Say what the term means before opening it. Then compare your explanation with the definition, notation, and example.
Inverse matrixAA^-1 = A^-1A = I
For an invertible square matrix A, the unique matrix that produces the identity when multiplied on either side of A.
How to read: A times A inverse equals the identity
Example: If Ax = b, then x = A^-1b when A^-1 exists.
Textbook reference: Chapter 9, Section 9.1 Defining the Inverse, printed pages 301-302Singular matrix
A square matrix that has no inverse, equivalently one whose determinant is zero.
Textbook reference: Chapter 9, Section 9.1 Defining the Inverse, printed page 302Nonsingular matrix
A square matrix that has an inverse, equivalently one whose determinant is nonzero.
Textbook reference: Chapter 9, Section 9.1 Defining the Inverse, printed page 302Determinant|A| or det(A)
A scalar calculated from a square matrix that indicates, among other things, whether the matrix is invertible.
How to read: the determinant of A
Example: For a 2 x 2 matrix, det(A) = a_11a_22 - a_12a_21.
Textbook reference: Chapter 9, Section 9.1 Defining the Inverse, printed page 305Geometric interpretation of a determinant
The absolute determinant measures the area or volume scaling generated by the matrix's column vectors, while its sign records orientation.
Example: For two vectors in ℝ², the parallelogram area is the absolute determinant of the matrix with those vectors as columns.
Textbook reference: Chapter 9, Section 9.1 Defining the Inverse, printed pages 315-316Linear dependence
A relationship in which at least one row or column vector can be expressed from the others, so the vectors do not provide fully distinct information.
Example: A matrix with one row equal to twice another has linearly dependent rows and determinant zero.
Textbook reference: Chapter 9, Section 9.1 Defining the Inverse, printed page 309MinorM_ij
The determinant of the submatrix obtained after deleting the row and column containing a specified entry.
How to read: minor M sub i j
Textbook reference: Chapter 9, Section 9.2 Obtaining the Determinant and Inverse of a 3 x 3 Matrix, printed pages 318-319CofactorC_ij = (-1)^(i+j) M_ij
A minor given the alternating sign determined by its row and column positions.
How to read: cofactor C sub i j
Textbook reference: Chapter 9, Section 9.2 Obtaining the Determinant and Inverse of a 3 x 3 Matrix, printed page 319Cofactor expansion
Computing a determinant by multiplying entries in one row or column by their cofactors and summing the products.
Textbook reference: Chapter 9, Section 9.2 Obtaining the Determinant and Inverse of a 3 x 3 Matrix, printed pages 319-320Cofactor matrixC = [C_ij]
The matrix formed by replacing every entry of a square matrix with its corresponding cofactor.
How to read: the cofactor matrix C
Textbook reference: Chapter 9, Section 9.2 Obtaining the Determinant and Inverse of a 3 x 3 Matrix, printed pages 320, 322Adjoint matrixadj(A) = C^T
The transpose of a matrix's cofactor matrix, used with the determinant to construct the inverse.
How to read: the adjoint of A
Textbook reference: Chapter 9, Section 9.2 Obtaining the Determinant and Inverse of a 3 x 3 Matrix, printed pages 320-321Matrix of minors[M_ij]
The matrix formed by replacing each entry of a square matrix with its associated minor.
How to read: the matrix of minors M sub i j
Textbook reference: Chapter 9, Section 9.3 The Inverse of an n x n Matrix and Its Properties, printed page 325Triangular matrix
A square matrix whose entries are all zero either below the main diagonal, above it, or both.
Example: The determinant of a triangular matrix is the product of its diagonal entries.
Textbook reference: Chapter 9, Section Chapter Review, printed page 344Cramer's rulex_j = det(A_j) / det(A)
A determinant method for solving a nonsingular system of n linear equations, one unknown at a time.
How to read: x sub j equals the determinant of A sub j divided by the determinant of A
Textbook reference: Chapter 9, Section 9.4 Cramer's Rule, printed pages 329-330Reduced-form equation
An equation expressing an endogenous variable only in terms of exogenous variables and fixed parameters.
Example: A reduced form for the interest rate removes income, consumption, and investment as simultaneous unknowns.
Textbook reference: Chapter 9, Section 9.4 Cramer's Rule, printed page 334Open Leontief input-output modelx = Ax + d
A model that finds the gross output each industry must produce to satisfy both industries' intermediate needs and an externally specified final demand.
How to read: gross output x equals intermediate demand Ax plus final demand d
Textbook reference: Chapter 9, Section 9.4 Cramer's Rule, printed pages 335, 337Production vector (gross output vector)x
A column vector listing the total output of every industry, including production used as intermediate inputs.
How to read: gross output vector x
Textbook reference: Chapter 9, Section 9.4 Cramer's Rule, printed pages 335, 337Final-demand vectord
A column vector listing demand for each industry's output from final users rather than from producing industries.
How to read: final-demand vector d
Textbook reference: Chapter 9, Section 9.4 Cramer's Rule, printed pages 335, 337Matrix of production coefficientsA = [a_ij]
The input-output matrix whose entry a_ij gives industry i output required per unit of industry j output.
How to read: A is the matrix with entry a sub i j
Textbook reference: Chapter 9, Section 9.4 Cramer's Rule, printed page 336Leontief inverse(I - A)^-1
The matrix that converts a final-demand vector into the gross output needed throughout an interdependent production system.
How to read: the inverse of I minus A
Example: Gross output is x = (I - A)^-1 d.
Textbook reference: Chapter 9, Section 9.4 Cramer's Rule, printed page 337Feasible final-demand vector
A final-demand plan whose implied gross output can be produced without exceeding the economy's available primary inputs.
Example: A demand vector is infeasible if the associated output plan requires more labor or capital than the economy has.
Textbook reference: Chapter 9, Section 9.4 Cramer's Rule, printed pages 339, 341End-of-chapter problems
Practice all 9 Chapter Review Exercises
The index covers every numbered Review Exercise in this chapter of Mathematics for Economics, 3rd edition. Each reference opens the textbook exercise itself, including every listed subpart, with MFE hints, a symbol guide, a worked explanation, progress tracking, and similar-problem practice.
Practice end-of-chapter problems