CHAPTER 2

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CHAPTER OUTLINE

2.1 Introduction

2.2 Types of Matrices

2.3 Determinants

2.4 The Inverse of a Square Matrix

2.5 Types of Solutions to Systems of Linear Equations

2.6 Solving Systems of Equations

2.7 Eigenvalues and Eigenvectors

2.8 Applications of Matrices

2.1 INTRODUCTION

Definition 2.1

A matrix is a rectangular array of elements or entries involving m rows and n columns.

Definition 2.2

i. Two matrices are said to be equal if m = r and n = s ( are same size ) then A = B, and corresponding elements throughout must also equal where

ii. If are called the main diagonal of matrix A.

ija

[ ] and [ ]ij m n ij r sA a M B b M

for all 1 and 1 .ij ija b i m j n

11, 22 33 for , then the entries , ,... ija i j a a a

Example 2.1 (Exercise 2.1 in TextBook):

Find the values for the variables so that the matrices in each exercise are equal.

i.

ii.

2 4 12

9 3 9

x y

z

3 12 5

2 8 6 8

x y

z

Example 2.2 (Exercise 2.2 in Textbook):

i. Give the order of each matrix.

ii. Identify or explain why identification is not possible.

1 54 7 5

If and B= 0 5 66 8 1

12 8 3

5

e

A

23 32 and a a

2.2 TYPES OF MATRICES

Square Matrix

A square matrix is any of order matrix and has the same number of columns as rows.

Diagonal Matrix

An matrix is called a diagonal matrix if

n n

n n0 and 0 for .ii ija a i j

1 1 24 5

, 3 5 76 7

2 1 4

A B

1 0 01 0

, 0 2 00 2

0 0 3

A B

Example 2.3 (Exercise 2.3 in Textbook):

Determine the matrices A and B are diagonal or not.

i.

ii.

2 0 0

0 2 0

0 0 5

A

6 0 0

0 2 0

0 0 5

B

Scalar Matrix

Scalar matrix- the diagonal elements are equal.

Identity Matrix

Identity matrix is called identity matrix with “1” on the main diagonal and “0”.

3 0 02 0

- scalar matrix, 0 5 0 -not a scalar matrix0 2

0 0 3

A B

1 0 01 0

- identity matrix, 0 1 0 -not an identity matrix0 1

0 0 3

A B

Zero Matrices

Zero Matrices – contain only “0” elements.

Negative Matrix

A negative matrix of

Upper Triangular

Upper triangular – if every element leading diagonal is zero.

is denoted by where .ij ijm n m nA a A A a

0 0

0 0A

3 1 3 1,

2 4 2 4A A

1 2 2 5 0 0

0 9 7 - Upper Triangular 6 2 0 - Lower Triangular

0 0 7 1 2 1

A B

Transpose Matrix

If is an matrix, then the transpose of , is

the matrix defined by .

TTij ij

T

ij ij

A a m n A A a

n m a a

2

i. 2 3 4 3

4

1 3 1 4ii.

4 2 3 2

T

T

A A

A

Properties of Transposition Operation

Let A, B matrices (different order) and k. Then

Example (Exercise 2.5 in Textbook):

i. ( )

ii. ( )

iii. ( )

T T

T T

T T T

A A

kA kA

A B A B

11 2 3

and 3 ,find4 5 2

2

i. (2 )

ii. ( )

T

T

B C

B

BC

Symmetric Matrix

Symmetric matrix – where the elements are obey the rule

Example (Exercise 2.6 in Textbook):

ij jia a

5 1 4

If 1 2 2 , find and show that it is symmetric matrix.

4 2 3

TA A

Skew Symmetric Matrix

Skew symmetric matrix –

Example (Exercise 2.7 in Textbook):

If , where elements obey the rule

, so that the leading diagonal must contain zeros.

0 1 3

Skew symmetric matrix of order 3 3 is 1 0 6

3 6 0

T

ij ji

A A

a a

0 1 4

If 1 0 7 , find and show that it is a skew

4 7 0

symmetric matrix.

TA A

Row Echelon Form (REF)

An matrix A is said to row echelon form (REF) if ot satisfies

the following properties:

i. Any rows consisting entirely of zeros occur at

the bottom of matrix.

ii. For each row that does not consist

m n

entirely of zeros, the first nonzero

entry is 1(called a leading 1)

iii. For each non zero row, the number 1 appears to the right of

the leading 1 of the previous row.

Example (Exercise 2.8 in Textbook):

Determine whether each matrix is in row echelon form.

1 2 1 2

i. 0 0 0 0 - not in row echelon form

0 1 2 4

1 2 3 4

ii. 0 2 1 2 not in row echelon form

0 0 1 3

1 0 3 4

iii. 0 1 9 2 - row echelon form

0 0 1 3

A

B

D

Reduced Row Echelon Form (RREF)

A matrix is said RREF if it satisfies the following properties:

Any rows consisting entirely of zeros occur at the bottom of the matrix.

For each row that does not consist entirely of zeros, the first nonzero entry is 1 (called a leading 1).

For each non zero row, the number 1 appears to the right of the leading 1 of the previous row.

If a column contains a leading 1, then all other entries in the column are zero.

Example:

Determine whether each matrix is in reduced row echelon form.

0 1 0

i. 1 0 0 - not RREF

0 0 1

1 0 0 4

0 1 0 5ii. B - RREF

0 0 1 6

0 0 0 0

1 5 3

iii. C 0 1 4 - not RREF

0 0 0

A

Addition and Subtraction of Matrix

• Two matrices can be addition and subtraction only if they are both in the same order.

Properties of Matrices Addition and Subtraction

i.

ii. ( ) ( )

iii. 0

iv. ( ) 0

A B B A

A B C A B C

A A

A A

Example (Addition) :

1.

2.

3.

5 1 6 3

7 3 2 1

2 1 4 1

2 1 65

3 0

t

t tt

5 06 2

1 01 1

2 1

Example (Subtraction):

2 4 6 0 2 3

If 1 3 5 , 7 8 4 , then find

9 10 11 2 1 1

A B A B

Scalar Multiplication

• Scalar multiplication is denoted

•

Properties of Scalar Multiplication

where [ ].ijA A a

i. ( )

ii. ( )

iii. ( ) ( )

iv.

v. 0 0

A B A B

A A A

A A

IA A

A

For product AB to exist, it must that , that is the number

of columns A must be the same as the number of rows B.

n p

Properties of Matrix Multiplication

i. ( ) (distributive law)

ii. ( ) (distributive law)

iii. ( ) ( ) (associative law)

iv. ( ) ( )

A B C AB AC

B C A BA CA

A BC AB C

AB A B

Example (Exercise 2.10 in Textbook):

1.

4 05 1 1 1

If 3 5 , and ,2 2 1 1

0 1

perform the indicated matrix operation. If an operation is not

defined, state the reason.

A B C

i. 4 3 ii. iii. ( )

iv. v. vi. ( )

B C BC CB A B C

A C B A A BC

Exercise:

1.

2.

3.

1 4

Given 5 5 , find

2 4

TA A

2 8 5Solve for x and y:

3 3 2 4 12

x y z

x z

4 1 2 4 1 1

5 3 3 6 3 1 5

1 0 4 12 9 1

4.

5.

6. Find

7 81 2 3

Given , 9 104 5 6

0 11

i. ii.

A B

AB BA

Find if 5,

2 4 6

1 3 5

9 10 11

B sA s

A

23 6 4

4

2.3 DETERMINANTS

Second Order Determinants

Determinant of A denoted by

Example :

Evaluate the determinant of

11 22 12 21

det A A

A a a a a

4 4 3 4,

5 1 11 2A B

Third Order Determinants

• Methods used to evaluate the determinants above is limited to only 2 X 2 and 3 X 3 matrices. Matrices with higher order can be solved using minor and cofactor methods.

Example (Exercise 2.11 in Textbook):

Evaluate the determinant of matrix

Exercise :

Find the determinants for matrix

3 0 0 3 1 0

2 1 5 , 3 4 0

2 5 1 1 3 5

A B

1 2 3 3 0 7

6 7 1 , 2 5 1

3 1 4 1 0 5

A B

Minors and Cofactors

Minor

Cofactor

Let 2 and [ ] .Matrices ( 1) ( 1) submatrix of

A is obtained by deleting the i-th row and j-th column of A

denoted by .

ij m n

ij

n A a n n

M

The cofactor of is defined as ( 1) .

The minor of is defined as .

i jij ij ij ij

ij ij

C a C M

a M

Example (Exercise 2.12 in Textbook) :

Find all the minors and cofactors of

Exercise :

0 2 1

3 1 2 .

4 0 1

A

3 1 4

2 5 6 .Find all the minor and cofactor.

1 4 8

Let A

High Order Determinants

1 1 2 21

1 1 2 2

If is cofactor of matrix A, then the determinant of matrix A can

be obtained by

Expanding along the i-th row

...

Expanding along the j-th column

...

ij

n

i i i i in in ij ijj

j j j j

C

A a C a C a C a C

A a C a C

1

n

nj nj ij iji

a C a C

Example ((Exercise 2.13 in Textbook) :

Find the determinant of

by expanding cofactors in the second row.

4 0 0 0 1 1 8 43 2 1

6 5 0 0 2 6 0 44 5 6 , and

1 3 1 0 2 0 2 62 3 1

1 2 7 3 0 2 8 0

A B C

Exercise:

Find the determinant of by expansion

the second column.

1 0 0 1

3 1 2 2

1 0 2 1

2 0 0 1

A

Adjoint

Example (Exercise 2.14 in Textbook) :

Find the adjoint of the matrix

Let is an matrix, then the tranpose of the matrix of cofactors

of A is called the matrix to adjoint to A, and it is denoted by

adj [A] .T

ij

A n n

C

2 9 4

7 6 0

6 7 6

A

Exercise :

Find the adjoint of the matrix

1.

2.

3 2 1

1 6 3 .

2 4 0

A

1 2 3

6 7 1 .

3 1 4

A

THE INVERSE OF A SQUARE MATRIX

Inverse of a Matrix

1

1 1

1

If A is a square matrix of order n and if there exists a matrix

(read "A inverse") such that

then is called the multiplicative inverse of A or, more simply,

the inverse of A.

A

AA A A I

A

Example (Exercise 2.15 of Textbook) :

2 1 3 1i. ,

5 3 5 2

2 11 2

ii. , 3 13 4

2 2

A B

A B

Finding the Inverse of a 2 by 2 Matrix

Theorem 1

1 1If , then .

The matrix A is invertible if and only if 0.

If 0, then A does not have multiplication inverse.

a b d bA A

c d c aA

ad bc

ad bc

Example (Exercise 2.16 of Textbook) :

1 1 3 1 3 6Find A and ,if it exists, given A= and .

2 2 6 12B B

Finding the Inverse of a 3 by 3 or Higher Matrix by Using Cofactor Method

Example (Exercise 2.17 Of Textbook) :

1 1 1 0 2 1

Find the inverse of 0 2 1 and 3 1 2 ,

2 3 0 4 0 1

if it is invertible.

A B

Finding the Inverse of a 3 by 3 or Higher Matrix by Elementary Row Operation

Characteristics of Elementary Row Operations (ERO)

Example (Exercise 2.18 of Textbook) :

3 7i.A=

2 5

1 3ii.

2 6

1 1 1

iii. C 4 5 0

0 1 3

1 0 1

iv. D 1 2 0

3 0 4

B

Exercise :

4 1 6

i. A 2 1 3

3 2 5

1 3 1

ii. 1 1 2

2 3 4

B

SYSTEMS OF LINEAR EQUATIONS

• Types of Solutions

• Solving Systems of Equations

• Eigenvalue and Eigenvectors

Types of Solutions

• System with Unique Solution (Independent)

i. A system which has unique solution.

ii. Can find the values of 1 2 3, ,x x x

1 6 14

0 1 4

1 2

2

1

1 2

6 14

4

14 6(4) 10

10, 4

x x

x

x

x x

• A system with Infinitely Many Solutions (Dependent)

i. A system has infinitely many solutions

ii. Row Echelon Form (REF) has a row of the form

iii. In general whatever value of , the equation is satisfied if . So we define a free variable, s.

0 0 0 0

1 2 4

0 0 0

1 24 2x x

2

1 4 2

x s

x s

• System with No Solution (Inconsistent)

i. A system has no solution.

ii. REF has a row of the form c is a constant. 0 0 0 c

1 2 4

0 0 3

Inverses of Matrices

When A is a square matrix. Note that A and B are matrices with numerical elements. To an expression for the unknowns, that is the elements of X.

Premultiplying both sides of the equation by the inverse of A, if it exists, to obtain

The left hand side can be simplified by noting that multiplying a matrix by its inverse gives the identity matrix, that is

Hence ,

Left hand side simplifies to

1 1A AX A B

1A A I 1 1( )A AX A A X IX

1, so form, IX X X A B

Example (Exercise 2.19 of Textbook):

Solve the following system of linear equations using the inverse matrix.

i.

ii.

1 2 3

1 2 3

1 3

6

2 3

3 0

x x x

x x x

x x

1 3

1 2

2 3

2 2 2

5 3 4

3 4 4

x x

x x

x x

Gaussian Elimination and Gauss-Jordan Elimination

• By using elementary row operations (ERO) on this matrix, matrix A will become reduced echelon form (REF) – Gaussian Elimination.

• Matrix A will become in reduced row echelon form (RREF) – Gauss-Jordan Elimination.

Example (Exercise 2.20 of Textbook):

Solve the system of linear equations using Gaussian elimination and Gauss-Jordan elimination.

i.

ii.

1 2 3

1 2 3

1 2 3

4 3 11

2 3 2 9

3

x x x

x x x

x x x

1 2 3

1 2 3

1 2 3

2 2 8

3 3 4

4 2 1

x x x

x x x

x x x

Cramer’s Rule

Example (Exercise 2.21 of Textbook):

i.

ii.

1 2

1 2

4 2 10

3 5 11

x x

x x

1 2 3

1 3

1 2 3

2 3 1

2 0

3 4 4 2

x x x

x x

x x x

EIGENVALUES AND EIGENVECTORS

Example (Exercise 2.22 of Textbook):

Find the eigenvalues and eigenvectors using Gaussian elimination method of the following matrices:

1 2 0

i. 1 1 1

3 2 2

4 0 1

ii. 2 1 0

2 0 1

A

B

Eigenvalues and eigenvectors can be computed as follows:

is the expansion of the determinant and polynomial p

APPLICATIONS OF MATRICES