Matrices

Matrices in Quantitative Economics

Overview of Matrix Algebra

• Important definitions and notation in matrix algebra:

  • Matrix: An array of numbers arranged in rows and columns.

  • Row: A horizontal set of elements in a matrix.

  • Column: A vertical set of elements in a matrix.

  • Vector: A matrix with a single row or single column.

  • Scalar: A single numerical value.

Key Objectives

• Understand the notation and terminology of matrix algebra.

• Perform operations on matrices:

  • Find the transpose of a matrix

  • Add, subtract, and multiply matrices

• Represent systems of linear equations in matrix notation.

• Determine matrix properties:

  • Identify singular or non-singular matrices

  • Calculate the determinant of a matrix

  • Compute the inverse of a 2x2 matrix

• Apply matrix operations in solving systems of linear equations and economic models.

Matrix Operations

Matrix Addition and Subtraction

• Matrices can be added or subtracted when they are compatible (same dimensions).

• Example:

  • A + B and A - B can be performed element-wise.

Matrix Multiplication

• Multiplying two matrices requires conformability (columns of the first matrix = rows of the second matrix).

• To multiply:

  • Multiply along the rows of the first matrix and down the columns of the second.

• Example:

  • For matrices A (m x n) and B (n x p), the resulting matrix C will be of dimension (m x p).

Transposition of a Matrix

• Transpose: Flipping a matrix over its diagonal.

  • Row becomes column and vice versa.

• Notation: If A is a matrix, its transpose is A^T.

Determinants and Inverses

Determinants

• Determinant: A unique scalar value associated with square matrices.

• For a 2x2 matrix A:

  A = [ \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} ]

  • The determinant is calculated as: det(A) = a_{11}a_{22} - a_{12}a_{21}.

• A matrix is invertible (has an inverse) if its determinant is non-zero.

Inverse of a Matrix

• Matrix Inverse: The matrix A^(-1) such that AA^(-1) = I (identity matrix).

• Calculating the inverse of a 2x2 matrix:

  1. If det(A) ≠ 0, exchange terms on the leading diagonal and change the signs of the off-diagonal terms.

  2. Divide each term by the determinant.

• [ A^{-1} = \frac{1}{det(A)} \begin{pmatrix} a_{22} & -a_{12} \ -a_{21} & a_{11} \end{pmatrix} ]

  • This only holds for non-singular matrices.

  • if the inverse is = to 0, then then state its singular and no inverse can be found

Cramer’s Rule

• Cramer’s Rule provides a method to solve systems of linear equations using determinants.

• Each variable can be solved as a ratio of determinants:

  • x_i = det(A_i)/det(A) where A_i is the obtained matrix by replacing the ith column of A by the constants on the right hand side of the equations.

Systems of Linear Equations

• Representing a system in matrix form:

  • Ax = b where A is the coefficient matrix, x vector of variables, and b is the constants vector.

• Solve for x using the inverse of A or Cramer’s rule.

Example Problem

• Given:

  • Ax = b,

  • Find inverse of A to deduce x.

  • Confirm calculations involving determinants to ensure a solution exists (det(A) ≠ 0).

Practical Applications

• Matrix operations in economics for solving models and systems balances:

  • Food and nutrition modeling: using matrices to ensure balanced dietary intake.

  • Economic model formulation: predicting outcomes based on linear relationships.

Resources for Further Learning

• Recommended reading:

  • Jacques Chapter 7 (excluding co-factors)

• Additional support: workshops, Q&A forums, peer study groups, consultation hours with instructors.