## Mathematics for Economists

## Chapter 8 : Matrices

### By Lund University

This chapter is an introduction to matrix algebra, or more generally, linear algebra. We will look at matrices and the algebra of matrices, that is, how to add and multiply two matrices and the rules that apply when we add and multiply matrices. In this chapter we also look at the transpose of a matrix, the inverse of a matrix and the determinant of a matrix. We also investigated the relationship between matrix algebra and systems of linear equations.

## Introduction to matrices

In the first section of this chapter we introduce matrices. We will then look at some special matrices such as diagonal matrices and the identity matrix. We then look at the row and column vectors and vectors in general.

#### Matrices, definitions

#### Special matrices

#### Row- and column vectors

#### Vectors

## Addition, scalar multiplication and transpose

We look at how to add to matrices and when you are allowed to add to matrices (same dimension). It is also possible to multiply and matrix by a scalar and we look at all the algebraic rules for matrix addition and scalar multiplication. In the final lecture we study the transpose of a matrix and related concepts such as symmetric matrices.

#### Matrix addition and scalar multiplication

#### Algebraic rules for matrix addition and scalar multiplication

#### Matrix transpose

## Matrices: Problems

Exercises on matrices

#### Problem: Construct matrix

#### Problem: Add matrices

#### Problem: Matrix equality

#### Problem: Add and subtract matrices

#### Problem: Linear combinations of 2 column vectors

#### Problem: Linear combinations of 3 row vectors

#### Problem: Linear combinations of 2 column vectors backwards

#### Problem: What can you say about the components of a vector?

#### Problem: Linear combination of 2 vectors, find the constants

#### Problem: Matrix transpose

#### Problem: Symmetric matrices

## Matrix multiplication

The fourth section is an entirely devoted to matrix multiplication. We look at under which conditions you are allowed to multiply two matrices, the exact method of how to multiply two matrices and all the rules that apply to matrix multiplication.

#### Dot product

#### Matrix multiplication, dimension

#### Matrix multiplication, computation

#### Matrix multiplication, rules

## Matrix multiplication: Problems

Exercises on matrix multiplication

#### Problem: Input and output vectors

#### Problem: Input and output vectors

#### Problem: Matrix products

#### Problem: Matrix algebra

#### Problem: Matrix algebra

#### Problem: The distributive law for matrices

#### Problem: Matrix products

#### Problem: Idempotent matrices

#### Problem: Transpose matrix expressions

#### Problem: Find matrix

## Matrix inverse, determinants and linear systems

We begin this section by looking at the inverse of a matrix and how to calculate the inverse of a 2 by 2 matrix. We then return to linear systems of equations and see that such systems can be written in matrix notation with a coefficent matrix. We also learn how the solution to a linear system, if it exists and is unique, is related to the inverse of the coefficient matrix. The final topic is determinants. For every square matrix we can calculate its determinant. The determinant is zero if and only if the matrix lacks an inverse.

#### Matrix Inverse

#### Linear systems of equations

#### Linear systems in matrix notation

#### Determinants

## Inverse, determinants and linear systems: Problems

Exercises on matrix inverse, determinants and linear systems