1 ln numerical methods - matrices and determinants

8/15/2019 1 LN Numerical Methods - Matrices and Determinants

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Numerical Methods

Matrices And Determinants



Matrices

Given by the expression

11 12 1

21 22 2

n

n

a a a

a a a A a

= =

L

L

where 1 ≤ i ≤ m (rows) and 1 ≤ j ≤ n (columns)Dimensions read as “m by n” (m × n)

1 2m m mna a a L



Types Of Matrices

1. Column or row vector

{ }

1

2

1 2

n

c

c R r r r C

= =

L

M

2. Null matrix nc 0 0 0

0 0 0

0 0 0

N

=

L

L

M M O M

L



3. Square matrix

Types Of Matrices11 12 1

21 22 2

1 2

n

n

n n nn

s s s

s s s

S

s s s

=

L

L

M M O M

L

4. Triangular matrix11 12 1

22 20

0 0

n

n

nn

u u u

u u

U

u

=

L

L

M M O M

L

11

21 22

1 2

0 0

0

n n nn

l

l l

L

l l l

=

L

L

M M O M

L



5. Diagonal matrix

Types Of Matrices

11

22

0 0

0 0

d

d D

=

L

L

M M O M

6. Identity matrix nnL

1 0 0

0 1 0

0 0 1

I

=

L

L

M M O M

L



7. Scalar matrix

Types Of Matrices1 0 0 0 0

0 1 0 0 0

k

k kI k

= =

L L

L L

M M O M M M O M

8. Transpose matrix

L L

11 21 1

12 22 2

1 2

m

mT

n n mn

a a a

a a a

A

a a a

=

L

L

M M O M

L



9. Symmetric matrix → if A = AT

10. Inverse matrix → A· A–1 = A–1· A = I

Types Of Matrices

11. Singular matrix → matrix that has no inverse

12. Orthogonal matrix → if AT = A–1, then A is an

orthogonal matrix.



13. Submatrix

Types Of Matrices

1 2 A A

A A A

=

{ }

{ } { }

11 12

1 2 13

21 22

3 31 32 4 33

,

,

a a A A a

a a

A a a A a

= =

= =



Operations1. Addition/Subtraction

→ dimensions should be congruent m = m B

ij ij ij

C A Bc a b

= ±

= ±

and n A = n B

Properties:

a. Commutative Property

b. Associative Property

2. Scalar multiplication ijkA ka=



Example

Given:

Evaluate:

Solution:

2 1 5 3,

3 4 0 2

3 4

2 1 5 3 6 3 20 123 4

A B

A B

− − = =

− −

−

− − − − − = −

Final Answer:

( )

( )

6 20 3 123 4

9 0 12 8

26 153 49 20

A B

A B

− − − −

− − − − − =

− − − −

− − = −



3. Matrix multiplication

Conditions1.Inner dimensions must conform, n A = m B

OperationsC A B= ⋅

Properties

1.Non-commutative AB ≠ BA

2.Associative A( BC ) = ( AB)C Algorithm ij ik kj

k

c a b= ∑



ExampleGiven:

Evaluate:

5 23 1 6

, 3 72 4 0

4 1

A B

AB

−

= = − − −

Final Answer:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )3 5 1 3 6 4 3 2 1 7 6 1

2 5 4 3 0 4 2 2 4 7 0 1

6 5

22 24

AB

AB

+ − − + − + − + = − + − + − − + +

− =

−



4. Elementary Row Operations → the following

are the only valid elementary row operations

a. Interchanging two rows, R1 ↔ R2

Operations

. ca ar mu t p cat on, 1 → 1

c. Row addition, R1 + R2 → R1’ or R2’



Determinants

Determinant → a scalar value obtained after

simplifying a square matrix

det or A A



Properties of Determinants1. The determinant of a matrix is the same to

that of its transpose.2. The determinant of a zero matrix is equal to

zero.

3. If a column/row is directly proportional toanother column/row, the determinant is equalto zero.

4. Switching two rows will switch the sign of itsdeterminant.



5. A scalar value k multiplied to a row will result to

the same value of determinant times k .6. Using valid elementary row addition/subtractionwill not change the value of the determinant.

Properties of Determinants

7. The determinant of the inverse of a matrix isequal to the reciprocal of its determinant:

8. The determinant of a triangular matrix is equal tothe product of its diagonal elements.

( ) ( )

1 1

detdet

A A

−=



Methods of DeterminantsA. Basket Method

Only valid for matrices with dimension n = 2

or n = 3.

( )

( ) ( ) ( )

11 12

11 22 21 12

21 22

11 12 13

21 22 23

31 32 33

11 22 33 12 23 31 13 21 32 31 22 13 32 23 11 33 21 12

, then detIf

If ,

then det

a a

A A a a a aa a

b b b

B b b b

b b b

B b b b b b b b b b b b b b b b b b b

= = −

=

= + + − + +



ExampleGiven

2 1 3

5 3 6

4 2 3

2 3 3 1 6 4 3 5 2

A

A

−

= −

− −

= − + − − + −

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

99 A

+ − − + − −

= −



B. Cofactor Expansion

An n – 1 reduction formula

involves following any row or column of a

Methods of Determinants

eterm nant an mu t p y ng eac e ement othe row or column by its cofactor. The sum of

these products equals the value of the

determinant.Cofactor of an element aij is ( ) ( )1 det

i j

ij ij A M

+

= −



ExampleGiven

choosing the 2nd row

2 1 3

5 3 6

4 2 3

A

−

= − − −

( ) ( ) ( )2 1 2 2 2 31 3 2 3 2 15 1 3 1 6 1

2 3 4 3 4 2

45 54 0

99

A

A

A

+ + +− − = − + − − − − − − −

= − − +

= −



C. Chio’s Condensation Method

An n – 1 reduction method using the first

element a11 as its reference


( )

11 13 11 111 12

21 23 21 221 22

11 12 11 13 11 1

2

11 31 32 31 33 31 3

11 12 11 12 11 1

1 2 1 3 1

n

n

nn

n

n

n n n n n nn

a a a aa a

a a a a a a

A a a a a a a a

a a a a a a

a a a a a a

−

=

L

L

M M O M

L



ExampleGiven

2 1 3

5 3 6

4 2 3

2 1 2 3

A

−

= −

− −

−

( )2 3 5 3 5 622 1 2 3

4 2 4 3

11 27199

0 182

A

A

− −

= −

− −

−= = −

−



D. Pivotal Method

An n – 1 reduction method selecting anyelement as its pivot element


or a × matr x, se ect ng a23 as t e p votelement:

( ) ( ) ( ) ( )

21 22

23 2321 22

23 23 21 22

23 23

11 12 13

2 3 2 3 11 13 12 13

23 23

31 13 32 13

31 32 33

1 1 1

a a

a aa aa a a a

a a

a a aa a a a

A a aa a a a

a a a

+ +

− −

= − = − − −



ExampleGiven

choosing element a32 as pivot element

2 1 3

5 3 6

4 2 3

A

−

= − − −

( )( )

( )( ) ( )( ) ( )( )

( ) ( ) ( )( )

( )( )

3 2

3

2

33 2 2

3

2

93 2 2

212

2 1 3

2 1 5 3 6

2 1

2 1 2 3 12 1

5 3 2 6 3

02 1 99

11

A

A

A

+

+

+

−

= − − − −

− − − − −= − −

− − − −

= − − = − −



E. Triangular Matrix Method

Converting a given matrix into an upper/lowertriangular matrix by means of valid elementary


.

The determinant of a triangular matrix is equal

to the product of its diagonal elements.



ExampleGiven 2 1 3

5 3 6

4 2 3

A

−

= − − −

( )( ) ( )( )

( )( )( )

1 12 1 2 3 1 32 2

2711 11

2 2 2

5 ' 4 '

2 1 3

0 , 2 9 99

0 0 9

R R R R R R

A A

+ − → + − →

−

= − = − = −

−



Inverse MatrixA. Generating the inverse matrix by using

Elementary Row Operations Augment the given matrix with an identity

, ,

augmented matrix to [ I : A–1].



ExampleGiven:

Solution:

2 1 3

5 3 6

4 2 3

A

−

= − − −

2 1 3 : 1 0 0−

( )( ) ( )( )1 12 1 2 3 1 32 2

27 511

2 2 2

:

4 2 3 : 0 0 1

5 ' 4 '

2 1 3 : 1 0 0

0 : 1 0

0 0 9 : 2 0 1

R R R R R R

−

− −

+ − → + − →

− − − − −



ExampleContinuing

( )( )21 2 111

6 6 211 11 11

27 5112 2 2

1 '

2 0 : 0

0 : 1 0

R R R+ →

− −

( )( ) ( )( )6 271 11 3 1 2 3 211 9 2 9

14 2 233 11 33

311 12 2 2

0 0 9 : 2 0 1

' '

2 0 0 :

0 0 : 1

0 0 9 : 2 0 1

R R R R R R

− −

+ − − → + − →

− − −



ExampleFinally, 1 2 1

1 1 2 2 3 32 11 9

7 1 133 11 33

31 211 11 11

2 1

' ' '

1 0 0 :

0 1 0 :

0 0 1 : 0

R R R R R R→ → − →

−

−

Therefore

7 1 133 11 33

1 31 211 11 11

2 1

9 90

A−

= − −



Inverse MatrixB. Generating the inverse matrix by using the

Adjoint Matrix and the determinant

( )1 Adj A

−=

where

( ) ( )cof

T

ij Adj A a =

A



ExampleGiven:

Solution:

2 1 3

5 3 6

4 2 3

A

−

= − − −

T − −

( )

2 3 4 3 4 2

1 3 2 3 2 199,

2 3 4 3 4 2

1 3 2 3 2 1

3 6 5 6 5 3

A Adj A

−

− − − − − − = − = − −

− − − −

− − −

− −



ExampleSimplifying further:

1

7 1 1

21 9 3

9 18 27

22 0 11

99 A

−

− − −

− − −

=−

Therefore,33 11 33

1 31 2

11 11 11

2 19 9

0

A−

= − −

1 ln numerical methods - matrices and determinants

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