Lecture 01 Matrices and Linear Equations

7/23/2019 Lecture 01 Matrices and Linear Equations

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Matrices

Linear Equations

Chapter One

Definition

• A matrix is a rectangular array of numbers.

The numbers in the array are called the

entries in the matrix.

• A matrix with size (order) m n is a matrix

with m rows and n columns.

Notation

• An entry that occur in row i and column j of a

matrix A will be denoted by aij

• Thus a general m n matrix might be written

2232221

1131211

m rows

n column

• A matrix can be denoted by an uppercase

letter such as A, B or C . A matrix can also be

denoted by [aij ],[bij ] or [cij ]. Therefore,

A = [aij ] =

2232221

1131211

• A matrix with n rows and n column is called a

square matrix of order n, and the entries a11,

a22, a33, …ann are main diagonal of A;

2232221

1131211

OPERATION OF MATRICES

Two matrices A = [aij] and B = [bij] are defined to be equal if they have thesame size and their corresponding entries are equal (aij = bij for all i and j).

Example 1:

32 A ,

x B ==> A = B only when x = 4

• Equal Matrices

Example 2:

The matrices

432121

are equal (A = B), if and only if w = -1, x = - 3, y = 0, and z = 5.

• Addition and Subtraction

If A = [aij] and B = [bij] are matrices of size m n, then their sum A+B isthe m n matrix given by adding the entries of B to the correspondingentries of A i.e.

A + B = [aij + bij].

Their differences A-B is the m n matrix obtained by subtracting theentries of B from the corresponding entries of A i.e.

A B = [aij bij]

Example 3:Let

52232526

A+C , B+C , AC , BC are undefined

If A = [aij] is an m n matrix and c is a scalar, the scalar multiple of A by c is the m n matrix given by

cA = c[aij]

The symbol - A represents the scalar product (1) A. Moreover, if A and B

are of the same size, then A B represents the sum of A and (1) B. That is,

A B = A + (1) B

• Sclar Multiples

Example 4:

421 A ,

720 B ,

101629

421 232 C B A

Properties

Let A, B and C be m n matrices and let c and d be scalars.

1. A + B = B + A Commutative Property of Matrix Addition

2. A + (B + C) = (A + B) + C Associative Property of Matrix Addition

3. (cd)A = c(dA) Associative Property of Scalar Multiplication

4. 1 A = A Scalar Identity

5. c(A + B) = cA + cB Distributive Property

6. (c + d)A = cA + dA Distributive Property

Matrix Multiplication

Definition: If A = [aij] is an m n matrix and and B = [bij] is an n p

matrix, the product AB is an m p matrix

AB = [cij] where

njin ji ji ji

kjik ij bababababac

...332211

In order for the product of two matrices to be defined, the number ocolumns of the first matrix must equal the number of rows of the second

matrix.

Example 5

cr bqap

ducr dt cqdscp

buar bt aqbsap AB

r q p B

Properties

Let A, B, and C be matrices and let c be a scalar.

1. A(BC) = (AB)C Associative Property of Multiplication

2. A(B + C)=AB + AC Distributive Property3. (A + B)C = AC + BC Distributive Property

4. c(AB) = (cA)B = A(cB) Distributive Property

Transpose Matrices

If A is an m n matrix, the transpose matrix T A , is the n m matrix whose

rows are the columns of A in the same order. In other words, the first row oT

A is the first column of A, the second row of T A is the second column of A,

and so on.

Example 6:

531T A

Properties

Let A and B denotes matrices of the same size, and let k denote a scalar.

1. If A is an m n matrix, then T A is an n m matrix

2. A AT T

3. T T kAkA

4. T T T B A B A

5.T T T A B AB )(

Symmetric Matrices

A matrix A is called symmetric if A = A

321T A ==> A is a symmetric matrix

Elementary Row Operations (ERO)

Interchange any two rows (row i and row j) and is denoted as

i R j R

Multiply row i by a scalar k(k 0) and is denoted as

i R ikR

Add multiple of row j to row i and is denoted as

a) ERO : Interchange rows 1 and 3

1 R 3 R

ERO : Multiply row 2 by 5, which can be read as

2 R becomes 5 X 2 R

2 R 25 R 5 5 5

Gaussian Elimination

Echelon form

A matrix satisfying the following conditions is said to be in reduced row-

echelon form (RREF):

1. If a row does not consist entirely of zeros, then the first nonzero number

in the row is a 1, which is called a leading 1.

2. If there are any rows that consist entirely of zeros, then they are grouped

together at the bottom of the matrix.

3. In any two successive rows that do not consist entirely of zeros, the

leading 1 in the lower row occurs farther to the right than the leading 1

in the higher row.4. Each column that contains a leading 1 has zeros elsewhere

Example

The following matrices are in reduced row-echelon form:

010001

70104001

How is about these…?

Example 7: Find x and y using

Gaussian Elimination Method for:

3/ 83/ 21

:formmatrixintoitChange

Solution

3/ 83/ 21

Example 8:Find x y and z using

Example 8:Find x, y and z using

Gaussian Elimination Method for:

3/53/53/100

3/43/13/21

3/53/53/100

3/53/43/10

3/43/13/21

)2(),1(

11223111

3/43/13/21

R R R R

1001)3(),4(

2301)15/1(

151500

)3/10(),3/2(

R R R R

Inverse of a 2x2 matrix

Lecture 01 Matrices and Linear Equations

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