### Introduction

In this Section we consider another direct method for obtaining the solution of systems of equations in the form $AX=B$ .

#### Prerequisites

- revise matrices and their use in systems of equations
- revise determinants

#### Learning Outcomes

- find an $LU$ decomposition of simple matrices and apply it to solve systems of equations
- determine when an $LU$ decomposition is unavailable and when it is possible to circumvent the problem

#### Contents

1 LU decomposition2 Using LU decomposition to solve systems of equations

3 Do matrices always have an LU decomposition?

3.1 Why not?

3.2 Can we get around this problem?