Lecture 01 Matrices and Linear Equations

7/23/2019 Lecture 01 Matrices and Linear Equations

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Matrices

&

Linear Equations

Chapter One

1



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.

2



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

as:

3

mnmmm

n

n

aaaa

aaaa

aaaa

321

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,

4

A = [aij ] =

mnmmm

n

n

aaaa

aaaa

aaaa

321

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;

5

nnnnn

n

n

aaaa

aaaa

aaaa

321

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:

94

32 A ,

9

32

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

6

• Equal Matrices



Example 2:

The matrices

540

432121

A

and

z y

xw

B

4

4221

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

7



• 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]

8



Example 3:Let

0724

4201

3012

A,

5423

1022

1534

B

,

012

321C

Then

5307

3221

4542

B A,

51141

52232526

B A

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

9



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

10

• Sclar Multiples



Example 4:

Let

062

421 A ,

531

720 B ,

1203

369C

11

41914

101629

1203

369 3

531

720

062

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

12



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

n

k

kjik ij bababababac

...332211

1

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.

13



Example 5

14

11

13

31.1

cr bqap

r

q

p

cba

12

13

32432

432

4

3

2

.2

f ed

cba

f ed

cba

32

3222

.3

ducr dt cqdscp

buar bt aqbsap AB

ut s

r q p B

d c

ba A



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

15



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:

685

073

421

A

604

872

531T A

16



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 )(

17



Symmetric Matrices

A matrix A is called symmetric if A = A

653

592

321

A

653

592

321T A ==> A is a symmetric matrix

18



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

i

R i j

R kR

19



a) ERO : Interchange rows 1 and 3

a b c

d e f

g h i

1 R 3 R

g h i

d e f

a b c

b)

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

2 R becomes 5 X 2 R

a b c

d e f

g h i

2 R 25 R 5 5 5

a b c

d e f

g h i

20



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

22



Example

The following matrices are in reduced row-echelon form:

00

00,

00000

00000

31000

10210

,

100

010001

,

1100

70104001

23



How is about these…?

10000

01100

06210

,

000

010

011

,

5100

2610

7341

24



25

Example 7: Find x and y using

Gaussian Elimination Method for:

1470

3/ 83/ 21

6

2116

3/ 83/ 21

3/ 1

2116

823

:formmatrixintoitChange

Solution

2116

823

12

1

R R

R

y x

y x

2,4

210

401

3/ 2

210

3/ 83/ 21

7/ 1

21

2

y x

R R

R

Example 8:Find x y and z using



Example 8:Find x, y and z using

Gaussian Elimination Method for:

26

3/53/53/100

5410

3/43/13/21

)3(

3/53/53/100

3/53/43/10

3/43/13/21

)2(),1(

11223111

3/43/13/21

)3/1(

1122

3111

4123

2

1312

1

R

R R R R

R

1,1,1

1100

1010

1001)3(),4(

1100

5410

2301)15/1(

151500

5410

2301

)3/10(),3/2(

3132

3

2321

z y x

R R R R

R

R R R R



Inverse of a 2x2 matrix

Lecture 01 Matrices and Linear Equations

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