Eva Apriyani 0902304

Laily Herni K 0902166

Noviawati 0902076

Rully Febrayanty 0902200

THE DETERMINANT

FUNCTION

DEFINITION

A permutation of the set of integers(1,2,. . .,n) is an arrangement of these integers in some

order without omissions or repetitions

There are six different permutations of the set of integers {1, 2, 3}. These are

EXAMPLE 1

)3,1,2( )2,1,3()3,2,1(

)1,3,2( )1,2,3()2,3,1(

EXAMPLE 2

Permutations of Four

Integers

List all permutations of the set of integers {1, 2, 3, 4}!

1

33

3

3

34

4

4

4

4

2

33

3

3

34

4

4

4

4

3

22

2

2

24

4

4

4

4

4

22

2

2

23

3

3

3

3

So , the solution of the problem :

1,2,3,4 2,1,3,4 3,1,2,4 4,1,2,3

1,2,4,3 2,1,4,3 3,1,4,2 4,1,3,2

1,3,2,4 2,3,1,4 3,2,1,4 4,2,1,3

1,3,4,2 2,3,4,1 3,2,4,1 4,2,3,1

1,4,2,3 2,4,1,3 3,4,1,2 4,3,1,2

1,4,3,2 2,4,3,1 3,4,2,1 4,3,2,1

We will denote a general permutation of the set (1,2,...,n) by (j1,j2,...,jn). Here, j1 is the first integer in

the permutation, j2 is the second, and so on. Aninversion is said to occur in a permutation (j1, j2, ...,

jn) whenever a larger integer precedes a smallerone. The total number of inversions occurring in a permutation can be obtained as follows: (1) find

the number of integersthat are less than j1 and that follow j1 in the

permutation; (2) find the number of integers thatare less than j2 and that follow j2 in the

permutation. Continue this counting process for j3, . . . ,jn-1 . The sum of these numbers will be thetotal number of inversions in the permutation.

(j1, j2, j3...,jn).

as the first integer in the permutation

as the second integer in the permutation

as the third integer in the permutation

inversion is said to occur in a permutation(j1, j2, ..., jn) whenever a larger integer precedes

a smaller one

Example 3Determine the number of inversion in the following permutation:

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

The number of inversion is

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

85 +0 + 1 + 1 + 1 =

(b) (2, 4, 1, 3)

The Number of inversion is

(2, 4, 1, 3)

3

(c) (1, 2, 3, 4)

1 + 2 + 0 =

There are zero inversion in this permutation

A permutation is called even if the total number of inversions is an even integer and is called odd if the total number of

inversions is an odd integer.

DEFINITION

Example 4

The following table classifies the various permutations of {1, 2, 3} as even or odd

Permutation Number of Inversion Classification

(1, 2, 3) 0 Even

(1, 3, 2) 1 Odd

(2, 1, 3) 1 Odd

(2, 3, 1) 2 Even

(3, 1, 2) 2 Even

(3, 2, 1) 3 Odd

Combinatorial Definition of The Determinant

By an elementary product from an n x m matrix A we shall mean any product of n entries from A. No two of which come from same row column

Example 5

List all elementary products from the matrices

(a)

2221

1211

aa

aa

Since each elementary product has two factors, and since each factor comes from a different row, an elementary product can be written in the form

Solution (a)

21aa

where the blanks designate column numbers. Since no two factors in the product come from the same column, the column numbers must be 12 or 21. Thus the only elementary products are a11 a22 and a12 a21 .

(b)

333231

232221

131211

aaa

aaa

aaa

Since each elementary product has three factors, each of which comes from a different row, an elementary product can be written in the form

Solution (b)

321aaa

Since no two factors in the product come from the same column, the column numbers have no repetitions; consequently, they must form a permutation of the set {1, 2, 3}. These 3! = 6 permutations yield the following list of elementary products

332211aaa

332112aaa

322113aaa

322311aaa

312312aaa

312213aaa

As this example points out, an n x n matrix A has n! elementary products. They are the products of the formwhere is a permutation of the set (1,2,…,n). By a signed elementary product from A we shall mean an elementary product multiplied by +1 or -1. We use the (+) if is an even permutation and the (-) ifis an odd permutation

jnjj naaa ...

21 21

njjj ,...,,

21

jnjj naaa ...

21 21

njjj ,...,,

21 njjj ,...,,

21

Example 6

List all signed elementary products from the

matrices

(i) (ii)

aa

aa

2221

1211

aaa

aaa

aaa

333231

232221

131211

Solution

(i)

Elementary

Product

Associated

PermutationEven or Odd

Signed

Elementary

Product

-

a11

a12

a 22

a 21

a11 a 22

a 21a12

(1,2)

(2,1)

even

odd

(ii) Elementary

Product

Associated

Permutation

Even or

Odd

Signed

Elementary

Product

(1, 2, 3) even

(1, 3, 2) odd -

(2, 1, 3) odd -

(2, 3, 1) even

(3, 1, 2) Even

(3, 2, 1) odd -

a11 a 22 a 33

a11 a 23a 32

a12

a12 a 23

a 21 a 33

a 31

a 13 a 21 a 32

a 22a 13 a 31

a11 a 22 a 33

a 23a 32

a 12a 21 a 33

a12 a 31a 23

a 21 a 32a 13

a 22a 13 a 31

a 11

Let A be a square matrix. We define det (A)

to be the sum of all signed elementary

products from .

DEFINITION

Example 7

a11 a12

a 21 a 22

= a11 a 22 - a12a 21

By multiplying the entries on the rightward arrow and

subtracting the product of the entries on the leftward

arrow

(i) det

(ii) deta11 a12 a 13

a 21 a 22

a 32 a 33a 31

a11 a12

a 32a 31

a 22a 21 =

a11a 22a 33 +

a 23

a12 a 31a 23 a 21 a 32a 13+ --

a 22a 13 a 31

a 23a 32a11 -

The determinant is then computed by summing the products

on the rightward arrows and subtracting the products on the

leftward arrows.

a12 a 21 a 33

EXAMPLE 8

Evaluating DeterminantsEvaluate the determinants of

Solution987

654

321

B

8

5

2

7

4

1

987

654

321

Using the earlier method, we get

det(B)=(45)+(84)+(96)-(105)-(-48)-(-72)=240

EVALUATING

DETERMINANTS BY

ROW REDUCTION

• Let A be a square matrix. If has a row of zeros or a

column of zeros, then det (A) = 0

a

a

a

aa

aa

aa

32

22

12

3332

2322

1312

0

0

0

0

0

0

000000

00

000

331232232213

231323123322

aaaaaa

aaaaaa

THEOREM 1

Proof:

Example 9

a

aa

aaa

aaaa

44

3433

242322

14131211

000

00

0

aaaa

aaa

aa

a

44434241

333231

2221

11

0

00

000

The general form of a lower triangle matrices 4×4

The general form of an upper triangle matrices 4×4

Calculate det (A), where

a

aa

aaa

aaaa

44

3433

242322

14131211

000

00

0

Example 10

det (A) = a11 a22…amn

Theorem 2

If A is triangle matrix nxn, then det (A) is the multiplied result of entries in leading diagonal , so det (A) = a11 a22…amn

Example 11

40000

89000

67600

15730

38372

1296)4)(9)(6)(3)(2(

Theorem 3

a. If B is the matrix that results when a single row or single column of A is multiplied by a scalar k , then det (B) = k det (A)

Proof:

let A = let B =

aaa

aaa

aaa

333231

232221

131211

333231

232221

131211

aaa

kakaka

aaa

= k

)det()det( BkA

333231

232221

131211

aaa

kakaka

aaa

aaa

aaa

aaa

333231

232221

131211

b. If B is the matrix that results when two rows or two columns of are interchanged, then det (B) = - det (A)

Proof: The first and second rows of A are interchangedaaa

aaa

aaa

333231

131211

232221

aaa

aaa

aaa

333231

232221

131211

)det()det( AB

c. If B is the matrix that results when a multiple of one row of A is added to another row or when a multiple of one column is added to another column, then det (B) = det (A)

A multiple of the second row of A is added to the first row.

proof :

aaa

aaa

kaakaakaa

333231

232221

231322212111

aaa

aaa

aaa

333231

232221

131211

)det()det( AB

121

410

321

A

121

410

1284

1A

121

321

410

2A121

232

321

3A

If we calculate this matrices using method on example 8 then we have

det (A) = -2, det (A1)= -8, det(A2) = 2, det (A3) = -2

Look at this :

det (A1)= 4 det(A), det(A2) = - det (A),

and det (A3) = det (A)

EXAMPLE 12

EXAMPLE 13

162

963

510

det A

162

510

963

162

510

321

3

5100

510

321

3

5500

510

321

3

100

510

321

)55)(3(

165)1)(55)(3(

the first and second row of A were interchanged

the common factor of 3 from the first row was taken through the determinant sign

-2 times the first row was added to the third row

-10 times the second row was added to the third row

a common factor of -55 from the last row was taken through the determinant sign

8411

5193

8462

4231

A

8411

5193

0000

4231

det A

Example 14

-2R1+R2

So, if square matrix have 2 rows which

comparable then determinant is zero

Next, we don’t reduced because from

theorem 1, we get that det(A)= 0

EXAMPLE 15

Every matrix have two rows which comparable so from examination, every matrix have determinant is zero.

82

41

872

423

872

151239

4185

2526

5413

THANK YOU