Chapter 8 · Test 2
Matrices for Economics
Matrix notation, operations, transposition, and special matrices.
What you will learn
- Multiply a matrix by a vector
- Read a transpose
- Determine product dimensions
Worked example
Multiply a matrix by a vector: follow the method step by step
Try the problem first, then compare your approach with the complete solution and the reasoning behind it.
Multiply a matrix by a vector
Let A = [[1, 2], [0, 3]] and v = [4, 5]ᵀ. If Av = [w₁, w₂]ᵀ, find w₁ + w₂.
wᵢ = row i of A · column v
In plain English: The ith entry of the product vector is found by taking the dot product of row i of matrix A with vector v.
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
- a 2 × 2 matrixHow to read: A
- v
- a 2 × 1 column vectorHow to read: v
- w = Av
- the resulting 2 × 1 vectorHow to read: w equals A times v
- i
- the row and output-component indexHow to read: i
Solution
- First row: w₁ = 1(4) + 2(5) = 4 + 10 = 14.
- Second row: w₂ = 0(4) + 3(5) = 15.
- Add the components: w₁ + w₂ = 14 + 15 = 29.
Logic and solving tips
- Each component of a matrix-vector product is a row-column dot product.
- Pair one matrix row with the vector, multiply corresponding entries, and add the products.
Where economists use it
An input-output analyst multiplies an industry's requirements matrix by planned final demand to estimate the total production needed from every supplying sector.
Key idea: Each component of a matrix-vector product is a row-column dot product.
Applied case study
Calculating direct requirements
Scenario
Economists multiply a direct-requirements matrix by an output vector to calculate industries' direct intermediate-input needs. Use the same transparent two-sector abstraction of BEA's requirements tables.
Problem
If A = [[0.2, 0.2], [0.1, 0.4]] and x = [100, 100]ᵀ billion dollars, find Ax.
Ax = [[0.2, 0.2], [0.1, 0.4]][100, 100]ᵀ
In plain English: Each output entry is formed by taking a row of the matrix and combining it with the corresponding entries of the vector.
Worked solution
- First row: 0.2(100) + 0.2(100) = 40.
- Second row: 0.1(100) + 0.4(100) = 50.
- Therefore Ax = [40, 50]ᵀ billion dollars.
- Check: x - Ax = [60, 50]ᵀ, the linked example's final-demand vector.
Economic interpretation: Producing $100 billion in each sector directly uses $40 billion of sector 1 output and $50 billion of sector 2 output as intermediate inputs.
Source: U.S. Bureau of Economic Analysis, Requirements Tables FAQ BEA supplies the definition and analytical use of direct requirements. The two-sector matrix and dollar values are illustrative.
Reasoning habits that help
- Each component of a matrix-vector product is a row-column dot product.
- Transposition reflects entries across the main diagonal by interchanging their indices.
- Matrix multiplication is conformable when the inner dimensions match, and the result takes the outer dimensions.
Where economists use these methods
- An input-output analyst multiplies an industry's requirements matrix by planned final demand to estimate the total production needed from every supplying sector.
- An econometrician transposes a data matrix when forming regression expressions such as XᵀX, which combine observations to estimate model coefficients.
- An econometrician checks dimensions before multiplying a data matrix by a coefficient vector so each observation receives one valid prediction instead of an invalid operation.
Chapter vocabulary
Study 29 key terms
Say what the term means before opening it. Then compare your explanation with the definition, notation, and example.
MatrixA = [a_ij]
A rectangular array of numbers whose row and column positions carry meaning and whose entries can be manipulated by the rules of matrix algebra.
How to read: matrix A with entry a sub i j
Example: A firm can place plants in rows, quarters in columns, and production values in the corresponding cells.
Textbook reference: Chapter 8, Section 8.1 General Notation, printed page 270Matrix entry (element)a_ij
A number in a specified row and column of a matrix.
How to read: a sub i j
Example: a_43 is the entry in row 4, column 3.
Textbook reference: Chapter 8, Section 8.1 General Notation, printed page 270Order (dimension) of a matrixm x n
The number of rows followed by the number of columns in a matrix.
How to read: m by n
Example: A 5 x 4 matrix has five rows and four columns.
Textbook reference: Chapter 8, Section 8.1 General Notation, printed page 270Row matrix (row vector)[v_1 v_2 ... v_n]
A matrix with one row, often used to list prices, coefficients, or another ordered set of values.
How to read: row vector v
Textbook reference: Chapter 8, Section 8.1 General Notation, printed pages 270-271Column matrix (column vector)v = [v_1, v_2, ..., v_n]^T
A matrix with one column, often used to represent quantities, prices, states, or unknowns.
How to read: column vector v
Textbook reference: Chapter 8, Section 8.1 General Notation, printed pages 270-271Vector
An ordered array with exactly one row or one column, treated as a special matrix.
Textbook reference: Chapter 8, Section 8.1 General Notation, printed page 271Matrix equalityA = B
Two matrices are equal only when they have the same dimensions and every pair of corresponding entries is equal.
How to read: matrix A equals matrix B
Textbook reference: Chapter 8, Section 8.1 General Notation, printed page 271Square matrixA is n x n
A matrix with the same number of rows and columns.
How to read: A is n by n
Textbook reference: Chapter 8, Section 8.1 General Notation, printed page 271Diagonal matrix
A square matrix whose off-diagonal entries are all zero.
Example: diag(2, 5) has 2 and 5 on its main diagonal and zeros elsewhere.
Textbook reference: Chapter 8, Section 8.1 General Notation, printed pages 271-272Identity matrixI_n
A diagonal matrix with ones on the main diagonal. Multiplying by it leaves a conformable matrix unchanged.
How to read: the n by n identity matrix
Example: I_3 A = A for every conformable matrix A.
Textbook reference: Chapter 8, Section 8.1 General Notation, printed page 272Null matrix (zero matrix)0
A matrix whose entries are all zero. It plays the additive role of zero in matrix algebra.
How to read: the zero matrix
Textbook reference: Chapter 8, Section 8.1 General Notation, printed page 272Input-requirements matrixA = [a_ij]
A matrix whose entry in row i and column j records how much output from industry i is needed per unit of industry j's output.
How to read: A is the matrix with entry a sub i j
Example: It summarizes how agriculture, mining, and manufacturing use one another's output as intermediate inputs.
Textbook reference: Chapter 8, Section 8.1 General Notation, printed pages 269-270Input-output model
A model of interdependent industries that connects each sector's production requirements with intermediate and final demand.
Textbook reference: Chapter 8, Section 8.1 General Notation, printed page 270Conformable for addition or subtraction
Having identical dimensions, so corresponding matrix entries can be added or subtracted.
Textbook reference: Chapter 8, Section 8.2 Basic Matrix Operations, printed page 273Matrix additionC = A + B, c_ij = a_ij + b_ij
Adding corresponding entries of two matrices with the same dimensions.
How to read: C equals A plus B
Example: Adding two half-year production matrices gives production for the full year.
Textbook reference: Chapter 8, Section 8.2 Basic Matrix Operations, printed pages 274-275Scalar multiplicationcA
Multiplying every entry of a matrix by the same real number.
How to read: scalar c times matrix A
Textbook reference: Chapter 8, Section 8.2 Basic Matrix Operations, printed pages 275-276Scalar
A single real number, distinguished from a vector or matrix.
Example: In 3A, the number 3 is the scalar multiplying every entry of A.
Textbook reference: Chapter 8, Section 8.2 Basic Matrix Operations, printed page 275Matrix equationAx = b
An equation written with matrices or vectors that compactly represents one or more scalar equations.
How to read: A times x equals b
Example: A system of simultaneous linear equations can be expressed as one matrix equation.
Textbook reference: Chapter 8, Section 8.2 Basic Matrix Operations, printed pages 276-277Matrix multiplicationC = AB, c_ij = sum_k a_ik b_kj
Forming each entry of a product by taking the inner product of a row from the first matrix with a column from the second.
How to read: C equals A times B
Example: Multiplying a row of quantities by a column of prices gives total expenditure.
Textbook reference: Chapter 8, Section 8.2 Basic Matrix Operations, printed pages 277, 280Product matrixC = AB
The matrix produced by multiplying two conformable matrices.
How to read: C is the product A B
Textbook reference: Chapter 8, Section 8.2 Basic Matrix Operations, printed page 280Conformable for multiplication
A pair of matrices for which the first matrix has as many columns as the second has rows.
Example: An m x n matrix times an n x q matrix produces an m x q matrix.
Textbook reference: Chapter 8, Section 8.2 Basic Matrix Operations, printed pages 277, 281Premultiplication and postmultiplication
Premultiplying B by A forms AB, while postmultiplying B by A forms BA. The order matters because these products can differ or one may be undefined.
Textbook reference: Chapter 8, Section 8.2 Basic Matrix Operations, printed page 281Matrix powerA^n
The product obtained by multiplying a square matrix by itself a stated number of times.
How to read: A to the nth power
Textbook reference: Chapter 8, Section 8.2 Basic Matrix Operations, printed page 283Transition matrixx^t = P^t x^0
A matrix that maps a current state vector into a later state vector, often by recording movement probabilities among states.
How to read: state x at time t equals P to the t times the initial state
Example: Labor-market transition probabilities can project employment, unemployment, and nonparticipation over time.
Textbook reference: Chapter 8, Section 8.2 Basic Matrix Operations, printed pages 282, 285TransposeA^T
The matrix formed by turning each row of the original matrix into the corresponding column.
How to read: A transpose
Textbook reference: Chapter 8, Section 8.3 Matrix Transposition, printed page 288Symmetric matrixA = A^T
A square matrix equal to its own transpose.
How to read: A equals A transpose
Textbook reference: Chapter 8, Section 8.3 Matrix Transposition, printed page 289Idempotent matrixA^2 = A
A square matrix that is unchanged when multiplied by itself.
How to read: A squared equals A
Example: Projection matrices used in statistics and econometrics are idempotent.
Textbook reference: Chapter 8, Section 8.4 Some Special Matrices, printed page 294Partitioned matrix
A matrix divided into smaller submatrices so a large system can be organized and manipulated in blocks.
Textbook reference: Chapter 8, Section 8.4 Some Special Matrices, printed page 295Trace of a matrixtr(A) = sum_i a_ii
The sum of the main-diagonal entries of a square matrix.
How to read: the trace of A
Textbook reference: Chapter 8, Section 8.4 Some Special Matrices, printed pages 295-296End-of-chapter problems
Practice all 8 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