8.2 Operations with Matrices

Video overview of all types of operations

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• Representation of Matrices

It is standard mathematical convention to represent matrices in any of the following three ways.

1. A matrix can be denoted by an uppercase letter such as A, B, or C.

2. 2. A matrix can be denoted by a representative element enclosed in brackets, such as [a_xy], [b_xy], or [c_xy]

3. A matrix can be denoted by a rectangular array of numbers such as the matrix in the picture

Equality of Matrices

1. 2 matricies A=[a_xy] and B=[b_xy] are Equal if they have the same order (mxn) and a_xy=b_xy

   1. BASICALLY SAYING THAT 2 matrices are equal if their corresponding entries are equal and have the same order so all entries are corresponding

Example 1 in book Equality of Matrices

Now onto Matrix Addition and Scalar Multiplication introduction

In this section, three basic matrix operations will be covered. The first two are matrix addition and scalar multiplication. With matrix addition, you can add two matrices (of the same order) by adding their corresponding entries.

• Definition of Matrix addition

if A=[a_xy] and B=[b_xy] are matrices of order MxN, their sum is the mxn matrix given by

A+B= [a_xy+b_xy]

THE SUM OF TWO MATRICES OF DIFFERENT ORDERS IS UNDEFINED

Example 2 in book Addition of Matrices

the one is undefined because A is of order 3×3 and B is of order 3×2

Scalar Multiplication introduction

in these operations, numbers usually referred to as scalars. scalars will always be real numbers in this book.

Scalar Multiplication Definition

Can multiply a matrix A by a scalar C by multiplying each entry in A by C

if A=[a_xy] is an mxn matrix and c is a scalar, the scalar multiple of A by C is the mxn matric given bycA=[ca_xy]

Matrix Subtraction (use -1 as scalar)

A-B represents the sum of A and (-1)B

A-B=A+(-1)B

Example 3 in book Scalar Multiplication and Matrix Subtraction

Helpful note Scalar Multiplication

Sometimes if fraction it can’t be convienant to rewrite the scalar multiple cA by factoring out C of every entry in matrix.

Properties of Matrix Addition and Scalar Multiplication

The properties of matrix addition and scalar multiplication are similar to those of addition and multiplication of real numbers.

Note that the Associative Property of Matrix Addition allows you to write expressions such as A+B+C without ambiguity because the same sum occurs no matter how the matrices are grouped. This same reasoning applies to sums of four or more matrices.

Example 4 in book Addition of More than Two Matrices

Example 5 in book Using the Distributive Property important note

Get same result if add 2 matrices first and then multiply the matrix by 3

Solving Matrix Equations Introduction

algebra of real numbers and algebra of matrices is very similar

Real numbers

x+a=b.

x+a-a=b+(-a)

x=b-a

MxN matrices

X+A=B

X+A+(-A)=B+(-A)

X=B-A

Example 6 in book Solving a Matrix Equation

• ACTUAL HARD PART AND VERY IMPORTANT TO KNOW COMING UP.

• MATRIX MULTIPLICATION

Matrix Multiplication DEFINITION ABOUT IT

If A=[a_xy] is an MxN matrix and B=[b_xy] is an NxP matrix

The product of AB is a MxP_ matrix

AB=[C_xy].

where c_xy=a_x1b_1y+a_x2b_2y+a_x3b_{3y+………..} a_xnb_ny

The definition of matrix multiplication indicates a row-by-column multiplication, where the entry in the xth row and yth column of the product AB is obtained by multiplying the entries in the x th row of A by the corresponding entries in the yth column of B and then adding the results.

• general PATTERN for matrix multiplication