### Matrix Algebra

#### Introduction

#### Matrix operations

#### Echelon matrices

#### Matrix properties

#### Matrix inverse

#### Matrix applications

#### Appendices

### Matrix Algebra: Table of Contents

#### Introduction

#### Matrix operations

#### Echelon matrices

#### Matrix properties

#### Matrix inverse

#### Matrix applications

#### Appendices

# Matrix Notation

Here, we describe how symbols are used on the Stat Trek web site to represent vectors, matrices, and other matrix algebra entities.

## Vectors

Bold-face, lower-case letters refer to vectors; and italic lower-case letters refer to vector elements. For example,

**a**and**x**refer to vectors**a**and**x**, respectively.*a*_{i}refers to the*i*th element in vector**a**.

## General Matrices

Bold-face, capital letters refer to matrices, italic capital letters refer to matrix elements, and subscripts reveal matrix dimension. For example,

**A**and**X**refer to matrices**A**and**X**, respectively.*A*_{i}_{j}refers to the element in row*i*and column*j*of matrix**A**.**A**_{i}_{j}refers to an*i*by*j*matrix**A**.

## Special Matrices and Matrix Properties

Special matrices are represented by special notation. For example,

**A'**refers to the transpose of matrix**A**.**I**refers to an identity matrix.**I**_{n}refers to an*n*x*n*identity matrix.**1**refers to the sum vector, a column vector having all of its elements equal to one.**1**_{n}is a 1 x n sum vector.- |
**A**| refers to the determinant of matrix**A**. **x**refers to a matrix of deviation scores derived from the raw scores of matrix**X**.

**Note**: Like vectors, deviation score matrices are often denoted
by a lower-case, boldface letter, such as **x**.
This can cause confusion, but usually the meaning is clear from the context.

## Elementary Operations

In many references, including this site, you will encounter a compact notation to describe elementary operations. That notation is shown below.

*R*means to interchange rows_{i}<--> R_{j}*i*and*j*of a matrix.*sR*means to multiply row_{i}--> R_{i}*i*by*s*.*sR*means to add_{i}+ R_{j}--> R_{j}*s*times row*i*to row*j*.*C*means to interchange columns_{i}<--> C_{j}*i*and*j**sC*means to multiply column_{i}--> C_{i}*i*by*s*.*sC*means to add_{i}+ C_{j}--> C_{j}*s*times column*i*to column*j*.

## Echelon Matrices

Matrix subscripts denote echelon forms.

**A**_{ref}denotes a row echelon form of matrix**A**.**A**_{rref}denotes the reduced row echelon form of matrix**A**.