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


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


Inverse, determinants and linear systems: Problems

Exercises on matrix inverse, determinants and linear systems

Problem: Determinants

Problem: Determinants

Problem: Matrix inverse

Problem: Matrix inverse

Problem: Matrix inverse

Problem: Matrix inverse and matrix transpose

Problem: Matrix equation

Problem: Matrix algebra

Problem: Linear equations

Problem: System of four equations

Problem: System of equations in matrix form