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# Matrices - Introduction. Lecture 1-3

## 1. Matrices

Introduction## 2. Matrices - Introduction

A set of mn numbers, arranged in a rectangular formation(array or table) having m rows and n columns and enclosed

by a square bracket [ ] is called mxn matrix (read “m by n

matrix”) .

1

1

4 2

3 0

a b

c d

The order or dimension of a matrix is the ordered pair having as

first component the number of rows and as second component the

number of columns in the matrix. If there are 3 rows and 2 columns

in a matrix, then its order is written as (3, 2) or (3 x 2) read as three

by two.

## 3. Matrices - Introduction

A matrix is denoted by a capital letter and the elements withinthe matrix are denoted by lower case letters

e.g. matrix [A] with elements aij

Amxn=

a11

a

21

am1

i goes from 1 to m

j goes from 1 to n

a12 ... aij

a22 ... aij

am 2

aij

ain

a2 n

amn

## 4. Matrices - Introduction

TYPES OF MATRICES1. Column matrix or vector:

The number of rows may be any integer but the number of

columns is always 1

1

4

2

1

3

a11

a21

am1

## 5. Matrices - Introduction

TYPES OF MATRICES2. Row matrix or vector

Any number of columns but only one row

1

1 6

a11

a12

0

3 5 2

a13 a1n

## 6. Matrices - Introduction

TYPES OF MATRICES3. Rectangular matrix

Contains more than one element and number of rows is not

equal to the number of columns

1 1

3 7

7 7

7 6

1 1 1 0 0

2 0 3 3 0

m n

## 7. Matrices - Introduction

TYPES OF MATRICES4. Square matrix

The number of rows is equal to the number of columns

(a square matrix A has an order of m)

mxm

1 1

3 0

1 1 1

9 9 0

6 6 1

The principal or main diagonal of a square matrix is composed of all

elements aij for which i=j

## 8. Matrices - Introduction

TYPES OF MATRICES5. Diagonal matrix

A square matrix where all the elements are zero except those on

the main diagonal

1 0 0

0 2 0

0 0 1

i.e. aij =0 for all i = j

aij = 0 for some or all i = j

3

0

0

0

0 0 0

3 0 0

0 5 0

0 0 9

## 9. Matrices - Introduction

TYPES OF MATRICES6. Unit or Identity matrix - I

A diagonal matrix with ones on the main diagonal

1

0

0

0

0 0 0

1 0 0

0 1 0

0 0 1

i.e. aij =0 for all i = j

aij = 1 for some or all i = j

1 0

0 1

aij

0

0

aij

## 10. Matrices - Introduction

TYPES OF MATRICES7. Null (zero) matrix - 0

All elements in the matrix are zero

0

0

0

aij 0

0 0 0

0 0 0

0 0 0

For all i,j

## 11. Matrices - Introduction

TYPES OF MATRICES8. Triangular matrix

A square matrix whose elements above or below the main

diagonal are all zero

1 0 0

2 1 0

5 2 3

1 0 0

2 1 0

5 2 3

1 8 9

0 1 6

0 0 3

## 12. Matrices - Introduction

TYPES OF MATRICES8a. Upper triangular matrix

A square matrix whose elements below the main

diagonal are all zero

aij

0

0

aij

aij

0

aij

aij

aij

i.e. aij = 0 for all i > j

1 8 7

0 1 8

0 0 3

1

0

0

0

7 4 4

1 7 4

0 7 8

0 0 3

## 13. Matrices - Introduction

TYPES OF MATRICES8b. Lower triangular matrix

A square matrix whose elements above the main diagonal are all

zero

aij

aij

aij

0

aij

aij

0

0

aij

i.e. aij = 0 for all i < j

1 0 0

2 1 0

5 2 3

## 14. Matrices – Introduction

TYPES OF MATRICES9. Scalar matrix

A diagonal matrix whose main diagonal elements are

equal to the same scalar

A scalar is defined as a single number or constant

aij

0

0

0

aij

0

0

0

aij

i.e. aij = 0 for all i = j

aij = a for all i = j

1 0 0

0 1 0

0 0 1

6

0

0

0

0 0 0

6 0 0

0 6 0

0 0 6

## 15. Matrices

Matrix Operations## 16. Matrices - Operations

EQUALITY OF MATRICESTwo matrices are said to be equal only when all

corresponding elements are equal

Therefore their size or dimensions are equal as well

A=

1 0 0

2 1 0

5 2 3

B=

1 0 0

2 1 0

5 2 3

A=B

## 17. Matrices - Operations

Some properties of equality:•IIf A = B, then B = A for all A and B

•IIf A = B, and B = C, then A = C for all A, B and C

A=

1 0 0

2 1 0

5 2 3

If A = B then

aij bij

b11 b12 b13

B=

b

b

b

21

22

23

b31 b32 b33

## 18. Matrices - Operations

ADDITION AND SUBTRACTION OF MATRICESThe sum or difference of two matrices, A and B of the same

size yields a matrix C of the same size

cij aij bij

Matrices of different sizes cannot be added or subtracted

## 19. Matrices - Operations

Commutative Law:A+B=B+A

Associative Law:

A + (B + C) = (A + B) + C = A + B + C

5 6 8

8 5

7 3 1 1

2 5 6 4 2 3 2 7 9

A

2x3

B

2x3

C

2x3

## 20. Matrices - Operations

A+0=0+A=AA + (-A) = 0 (where –A is the matrix composed of –aij as elements)

6 4 2 1 2 0 5 2 2

3 2 7 1 0 8 2 2 1

## 21. Matrices - Operations

SCALAR MULTIPLICATION OF MATRICESMatrices can be multiplied by a scalar (constant or single

element)

Let k be a scalar quantity; then

kA = Ak

Ex. If k=4 and

3 1

2 1

A

2 3

4 1

## 22. Matrices - Operations

3 1 3 112 4

2 1 2 1

8 4

4

4

2 3 2 3

8 12

4 1 4 1

16 4

Properties:

• k (A + B) = kA + kB

• (k + g)A = kA + gA

• k(AB) = (kA)B = A(k)B

• k(gA) = (kg)A

## 23. Matrices - Operations

MULTIPLICATION OF MATRICESThe product of two matrices is another matrix

Two matrices A and B must be conformable for multiplication to

be possible

i.e. the number of columns of A must equal the number of rows

of B

Example.

A

(1x3)

x

B =

(3x1)

C

(1x1)

## 24. Matrices - Operations

B xA

=

Not possible!

(2x1) (4x2)

A

x

B

=

Not possible!

(6x2) (6x3)

Example

A

(2x3)

x

B

(3x2)

=

C

(2x2)

## 25. Matrices - Operations

a11 a12a

21 a22

b11 b12

a13

c11 c12

b

b

21

22

a23

c21 c22

b31 b32

(a11 b11 ) (a12 b21 ) (a13 b31 ) c11

(a11 b12 ) (a12 b22 ) (a13 b32 ) c12

(a21 b11 ) (a22 b21 ) (a23 b31 ) c21

(a21 b12 ) (a22 b22 ) (a23 b32 ) c22

Successive multiplication of row i of A with column j of

B – row by column multiplication

## 26. Matrices - Operations

4 81 2 3

(1 4) (2 6) (3 5) (1 8) (2 2) (3 3)

4 2 7 6 2 (4 4) (2 6) (7 5) (4 8) (2 2) (7 3)

5 3

31 21

63 57

Remember also:

IA = A

1 0 31 21

0 1 63 57

31 21

63 57

## 27. Matrices - Operations

Assuming that matrices A, B and C are conformable forthe operations indicated, the following are true:

1. AI = IA = A

2. A(BC) = (AB)C = ABC -

(associative law)

3. A(B+C) = AB + AC - (first distributive law)

4. (A+B)C = AC + BC - (second distributive law)

Caution!

1. AB not generally equal to BA, BA may not be conformable

2. If AB = 0, neither A nor B necessarily = 0

3. If AB = AC, B not necessarily = C

## 28. Matrices - Operations

AB not generally equal to BA, BA may not be conformable1 2

T

5

0

3 4

S

0

2

1 2 3

TS

5 0 0

3 4 1

ST

0 2 5

4 3

2 15

2 23

0 10

8

20

6

0

## 29. Matrices - Operations

If AB = 0, neither A nor B necessarily = 03 0 0

1 1 2

0 0 2 3 0 0

## 30. Matrices - Operations

TRANSPOSE OF A MATRIXIf :

2 4 7

A 2 A

2x3

5 3 1

3

Then transpose of A, denoted AT is:

3T

A 2 A

T

2 5

4 3

7 1

aij a

T

ji

For all i and j

## 31. Matrices - Operations

To transpose:Interchange rows and columns

The dimensions of AT are the reverse of the dimensions of A

2 4 7

A 2 A

5

3

1

3

T2

A 3 A

T

2 5

4 3

7 1

2x3

3x2

## 32. Matrices - Operations

Properties of transposed matrices:1. (A+B)T = AT + BT

2. (AB)T = BT AT

3. (kA)T = kAT

4. (AT)T = A

## 33. Matrices - Operations

1. (A+B)T = AT + BT5 6 8

8 5

7 3 1 1

2 5 6 4 2 3 2 7 9

2 1 4 8 2

7

3 5 5 2 8 7

1 6 6 3 5 9

8 2

8 7

5 9

## 34. Matrices - Operations

(AB)T = BT AT1

1 1 0 2

0 2 3 1 8 2 8

2

1 0

1 1 2 1 2 2 8

0 3

## 35. Matrices - Operations

SYMMETRIC MATRICESA Square matrix is symmetric if it is equal to its

transpose:

A = AT

a b

A

b d

a b

T

A

b

d

## 36. Matrices - Operations

When the original matrix is square, transposition does notaffect the elements of the main diagonal

a b

A

c

d

a c

T

A

b

d

The identity matrix, I, a diagonal matrix D, and a scalar matrix, K,

are equal to their transpose since the diagonal is unaffected.

## 37. Matrices - Operations

INVERSE OF A MATRIXConsider a scalar k. The inverse is the reciprocal or division of 1

by the scalar.

Example:

k=7

the inverse of k or k-1 = 1/k = 1/7

Division of matrices is not defined since there may be AB = AC

while B = C

Instead matrix inversion is used.

The inverse of a square matrix, A, if it exists, is the unique matrix

A-1 where:

AA-1 = A-1 A = I

## 38. Matrices - Operations

Example:3

A 2 A

2

1

1

A

2

2

Because:

1

1

1

3

1 1 3 1 1

2 3 2 1 0

3 1 1 1 1

2 1 2 3 0

0

1

0

1

## 39. Matrices - Operations

Properties of the inverse:( AB) 1 B 1 A 1

1 1

(A ) A

T 1

1 T

(A ) (A )

1 1

1

(kA) A

k

A square matrix that has an inverse is called a nonsingular matrix

A matrix that does not have an inverse is called a singular matrix

Square matrices have inverses except when the determinant is zero

When the determinant of a matrix is zero the matrix is singular

## 40. Matrices - Operations

DETERMINANT OF A MATRIXTo compute the inverse of a matrix, the determinant is required

Each square matrix A has a unit scalar value called the

determinant of A, denoted by det A or |A|

If

then

1

A

6

1

A

6

2

5

2

5

## 41. Matrices - Operations

If A = [A] is a single element (1x1), then the determinant isdefined as the value of the element

Then |A| =det A = a11

If A is (n x n), its determinant may be defined in terms of order

(n-1) or less.

## 42. Matrices - Operations

MINORSIf A is an n x n matrix and one row and one column are deleted,

the resulting matrix is an (n-1) x (n-1) submatrix of A.

The determinant of such a submatrix is called a minor of A and

is designated by mij , where i and j correspond to the deleted

row and column, respectively.

mij is the minor of the element aij in A.

## 43. Matrices - Operations

eg.a11 a12

A a21 a22

a31 a32

a13

a23

a33

Each element in A has a minor

Delete first row and column from A .

The determinant of the remaining 2 x 2 submatrix is the minor

of a11

m11

a22

a23

a32

a33

## 44. Matrices - Operations

Therefore the minor of a12 is:m12

a21 a23

a31 a33

And the minor for a13 is:

m13

a21 a22

a31 a32

## 45. Matrices - Operations

COFACTORSThe cofactor Cij of an element aij is defined as:

Cij ( 1)i j mij

When the sum of a row number i and column j is even, cij = mij and

when i+j is odd, cij =-mij

c11 (i 1, j 1) ( 1)1 1 m11 m11

1 2

m12 m12

1 3

m13 m13

c12 (i 1, j 2) ( 1)

c13 (i 1, j 3) ( 1)

## 46. Matrices - Operations

DETERMINANTS CONTINUEDThe determinant of an n x n matrix A can now be defined as

A det A a11c11 a12c12 a1nc1n

The determinant of A is therefore the sum of the products of the

elements of the first row of A and their corresponding cofactors.

(It is possible to define |A| in terms of any other row or column

but for simplicity, the first row only is used)

## 47. Matrices - Operations

Therefore the 2 x 2 matrix :a11 a12

A

a21 a22

Has cofactors :

c11 m11 a22 a22

And:

c12 m12 a21 a21

And the determinant of A is:

A a11c11 a12c12 a11a22 a12a21

## 48. Matrices - Operations

Example 1:3 1

A

1

2

A (3)(2) (1)(1) 5

## 49. Matrices - Operations

For a 3 x 3 matrix:a11 a12

A a21 a22

a31 a32

a13

a23

a33

The cofactors of the first row are:

c11

a22

a23

a32

a33

a22a33 a23a32

a21 a23

c12

(a21a33 a23a31 )

a31 a33

a21 a22

c13

a21a32 a22a31

a31 a32

## 50. Matrices - Operations

The determinant of a matrix A is:A a11c11 a12c12 a11a22 a12a21

Which by substituting for the cofactors in this case is:

A a11(a22a33 a23a32 ) a12 (a21a33 a23a31) a13 (a21a32 a22a31)

## 51. Matrices - Operations

Example 2:1 0 1

A 0 2 3

1 0 1

A a11(a22a33 a23a32 ) a12 (a21a33 a23a31) a13 (a21a32 a22a31)

A (1)(2 0) (0)(0 3) (1)(0 2) 4

## 52. Matrices - Operations

ADJOINT MATRICESA cofactor matrix C of a matrix A is the square matrix of the same

order as A in which each element aij is replaced by its cofactor cij .

Example:

If

1 2

A

3 4

The cofactor C of A is

4 3

C

2

1

## 53. Matrices - Operations

The adjoint matrix of A, denoted by adj A, is the transpose of itscofactor matrix

adjA C

T

It can be shown that:

A(adj A) = (adjA) A = |A| I

Example:

1 2

A

3

4

A (1)( 4) (2)( 3) 10

4 2

adjA C

3

1

T

## 54. Matrices - Operations

1 2 4 2 10 0A(adjA)

10 I

3 4 3 1 0 10

4 2 1 2 10 0

(adjA) A

10 I

3 1 3 4 0 10

## 55. Matrices - Operations

USING THE ADJOINT MATRIX IN MATRIX INVERSIONSince

AA-1 = A-1 A = I

and

A(adj A) = (adjA) A = |A| I

then

adjA

A

A

1

## 56. Matrices - Operations

ExampleA=

1 2

3 4

1 4 2 0.4 0.2

A

10 3 1 0.3 0.1

1

To check

AA-1 = A-1 A = I

1 2 0 . 4 0 .2 1

AA

3 4 0.3 0 .1 0

0 .4 0 . 2 1 2 1

1

A A

0.3 0.1 3 4 0

1

0

I

1

0

I

1

## 57. Matrices - Operations

Example 23 1 1

A 2 1 0

1 2 1

The determinant of A is

|A| = (3)(-1-0)-(-1)(-2-0)+(1)(4-1) = -2

The elements of the cofactor matrix are

c11 ( 1),

c12 ( 2),

c13 (3),

c21 ( 1),

c22 ( 4),

c23 (7),

c31 ( 1),

c32 ( 2),

c33 (5),

## 58. Matrices - Operations

The cofactor matrix is therefore3

1 2

C 1 4 7

1 2

5

so

and

1 1 1

adjA C T 2 4 2

3 7 5

1 1 1 0.5 0.5 0.5

adjA 1

1.0 2.0 1.0

1

A

2

4

2

A

2

3 7 5 1.5 3.5 2.5

## 59. Matrices - Operations

The result can be checked usingAA-1 = A-1 A = I

The determinant of a matrix must not be zero for the inverse to

exist as there will not be a solution

Nonsingular matrices have non-zero determinants

Singular matrices have zero determinants