Math9

Országok listájaHungaryBudapesti Corvinus EgyetemGazdálkodástudományi KarNemzetközi gazdálkodás (angol nyelven)Mathematics1JegyzetekMath9

Determinants
9.1 On the parity of permutations

Denition 9.1 Let H be a nonempty set. A bijective map : H - H is said to be a permutation of the elements of H. Remark 9.1 If H is an n element set say H = {1, 2, . . . , n} then the number of · permutations dened on H is n! (read n factorial, that is 12n. The set of all permutations of the set {1, 2, . . . , n} is denoted by Sn . A permutation : H H can be represented by a 2 × n table of the form = i1 i2 . . . in , where ij is ... n 1 2 the element of H corresponded to i by the map . Denition 9.2 Let =
1 i1 2 i2 ... n . . . in (ij , ik ) is

{1, 2, . . . , n}. A pair of elements if j < k, but ij > ik.

be a permutation of the set H = called an inversion in permutation
1 5 2 1 3 4 4 2 5 3

For instance, the number of inversions in the permutation
6, namely (5, 1), (5, 4), (5, 2), (5, 3), (4, 2) and (4, 3).

is

Denition 9.3 A permutation is said to be even if the number of its inversions is even number, otherwise it is an odd permutation. Denition 9.4 The function sgn : {-1, 1} dened on the set of permutations as follows: 1 if is even, sgn() = -1 if is odd.
9.2 Determinants

Denition 9.5 Let A = [aij ] Rn×n be a square matrix. Its determinant det A denoted by
a11 a21

. . .

a12 a22

. . .

... ... ...

a1n a2n

. . .

an1

an2

ann

equals
sgn()a1(1) a2(2) · · · an(n) .
Sn

n

is called the order of the determinant.
1

9.2.

DETERMINANTS

Remark 9.2 The determinant of a matrix of order n is the sum of n! terms, where each term is a product of n factors. The factors of a term is entries of the matrix chosen exactly one from each row and each column. The sign of the term is the sign of the permutation that maps the row indices onto the column indices.
To calculate the determinant using the denition is getting more dicult as the order is increasing. For instance, the determinant of a matrix of order 4 is the sum of 24 terms. On the other hand if the matrix has a special form, then to nd its determinant can be very simple. For example, if the matrix is an upper triangle matrix, then
a11 0

. . .

a12 a22

. . .

... ... ...

a1n a2n

. . .

= a11 a22 . . . ann .

0

0

ann

Example 9.1 Determine the determinant of matrices A = [aij ] of order n = 1, 2, 3.
n = 1 case: det(a) = a. n = 2 case: det A = n = 3 case: a11 a21 a31 a12 a22 a32 a13 a23 a33 = a11 a22 a33 + a12 a23 a31 + a13 a21 a32 -a11 a23 a32 - a12 a21 a33 - a13 a22 a31 a11 a21 a12 a22 = a11 a22 - a12 a21

Example 9.2 By the general rule nd the following determinant:

1. 2.
1 4 1 2 1 2 3 2 4

4 3

2 5

= 4 · 5 - 2 · 3 = 14

=

1·1·4+2·2·1+3·4·2-

- 3 · 1 · 1 - 1 · 2 · 2 - 2 · 4 · 4 = -7

Proposition 9.1 (Properties of determinants) zero, then its determinant is 0.

1. If each entry of a matrix A is

2. The determinant of a matrix equals the determinant of its transpose: |A| = A . 3. If two rows of a matrix A is interchanged, then the sign of its determinant is changing. 4. The determinant of a matrix A having two equal rows is zero.

2

9.2.

DETERMINANTS

5. If each entry of the ith row of the matrix A is the sum of two terms, then its determinant is the sum of two determinants, namely:
a11

. . .

a12

. . .

··· ··· ··· a11

a1n

. . .

det A = bi1 + ci1

. . .

bi2 + ci2

. . .

bin + cin

. . .

=

an1 a11

an2 a1n

ann a12

. . .

a12

. . .

··· ··· ···

. . .

. . .

. . .

··· ··· ···

a1n

. . .

=

bi1

. . .

bi2

. . .

bin

. . .

+

ci1

. . .

ci2

. . .

cin

. . .

an1

an2

ann

an1

an2

ann

Notice that the entries which are not in the ith row are the same in both determinants as in the original one. 6. If each entry of a row of the matrix A has a common factor, then it can be factored out from its determinant, more accurately:
a11

. . .

a12

. . .

··· ··· ··· ··· ··· ···

a1n

. . .

det A= · ai1

. . .

· ai2

. . .

· ain

. . .

=

an1 a11

an2

ann a1n

. . .

a12

. . .

. . .

=·

ai1

. . .

ai2

. . .

ain

. . .

.

an1

an2

ann

7. If a row of the matrix A is a scalar multiple of its another row then its determinant is zero. 8. If a scalar multiple of a row of A is added to its another row then its determinant is not changing.
Remark 9.3 The properties listed above with respect to the rows of a determininant hold true for its columns as well. Example 9.3 Using the above properties nd the following determinants :
1 2 1 3 1 4 2 2 1 5 1 0 1 1 . -1 5

3

9.2.

DETERMINANTS

Solution:

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

1 1 -1 5

=

1 0 0 0

1 1 1 2 3 -1 1 0 -2 -1 -3 2

1 0 =- 0 0

1 1 1 1 0 -2 2 3 -1 -1 -3 2 1 -2 3 3

=

1 1 0 1 =- 0 0 0 0

1 1 0 -2 3 3 -3 0

1 1 1 0 1 0 =- 0 0 3 0 0 0

= -9

First (-2) times, (-1) times and (-3) times the st row was added to the 2nd, 3rd and 4th row respectively to achieve that the 2nd, 3rd and 4th entry in the rst

column become zero. Then we interchanged the second and third row therefore the determinant changed its sign. Using similar row operations the third and fourth entry of the second column and the fourth entry of the third column could be converted to zero to obtain an upper triangular matrix. Finally we used the fact that the determinant of an upper triangular matrix is the product of the diagonal entries.

Remark 9.4 In the previous example using elementary row operations we converted the given matrix to an upper triangular one and its determinant is simple the product of the entries in the main diagonal.
Let us summarize the elementary row/column operations and their eect to the determinant of the matrix:

Denition 9.6 The elementary row/column operation of matrices are:
-

Interchanging two rows or columns; determinant changes its sign, Adding a multiple of one row or column to another; determinant is unchanged, Multiplying any row or column by a nonzero scalar; determinant is multiplied by the scalar.

Theorem 9.2 Using elementary row operations any square matrix can be converted into an upper triangular matrix.
. . . Let A = ai1 . . . an1 a11 a12

. . .

... ... ...

a1n



Proof:

ai2

. . .

1st step: If each entry of the rst column is zero, then there is nothing to do with the 1st column. Otherwise interchanging rows a nonzero entry in the rst column ai1 can be taken to the rst row. Assuming that a11 = 0 adding - a11 times the rst row to the ith one for each i = 2, . . . , n, it can be achived that each entry in the rst column except the rst one becomes zero. 2nd step: We carry out the operations of the 1st step on the n - 1 × n - 1 matrix block obtained from the converted one by omitting both the rst row and rst column. (The rst entries of the 2nd . . . nth rows are already zeros, thus they do 4

an2

ain be an arbitrary square matrix. . . . ann

. . .

9.3.

DETERMINANT EXPANSION BY MINORS

not change for the elementary row operations.) Similarly continued on the n - 2 × n - 2 and so on 2 × 2 matrix block each entry under the diagonal in the converted matrix becomes zero.

Theorem 9.3 The determinant of a matrix is zero if and only if it is singular. Theorem 9.4 (Determinant of matrix product) Let A and B be matrices of the same order. Then
|A · B| = |A| · |B| .

It immediately follows from the denition of the determinant, that the determinant of the identity matrix of any order is 1. Therefore it follows from the previous theorem and the properties of the determinant that

Theorem 9.5 The determinant of a square matrix A is zero if and only if A is singular. Proposition 9.6 If A is a regular matrix, then
det A-1 = 1 . det A

Denition 9.7 Two matrices A and B are said to be similar if there exists an invertible matrix C, such that B = C-1 AC. Proposition 9.7 The determinants of similar matices are equal to each other.
9.3 Determinant expansion by minors

Let us consider again the determinant of a matrix of order 3
|A| = a11 a21 a31 a12 a22 a32 a13 a23 a33 =

=

a11 a22 a33 + a12 a23 a31 + a13 a21 a32 -a11 a23 a32 - a12 a21 a33 - a13 a22 a31

At rst sight it seems to be dicult but it becomes more simple using the concepts of minors. Notice that the determinant can be written in the form
|A| = a11 (a22 a33 - a23 a32 ) - a12 (a21 a33 - a23 a31 ) + +a13 (a21 a32 - a22 a31 ) ,

where determinants of order 2 appears as factors:
|A| = a11 +a13 a22 a32 a21 a31 a23 a33 a22 a32 - a12 . a21 a31 a23 a33 +

Thus, to determine the determinant of order 3 can be led back to the calculation of determinants of order 2. In general to nd the determinant can be led back to the calculation of determinants having smaller order. 5

9.3.

DETERMINANT EXPANSION BY MINORS

Denition 9.8 If k rows and k columns is omitted from a matrix, then the determinant of the obtained matrix is said to be a minor of the determinant of the original matrix. let A = [aij ] be n × n matrix and denote by Aij the matrix that can be obtained from A by omitting its ith row and j th column. The determinant of the (n - 1) × (n - 1) matrix Aij is called the minor Mij corresponding to the entry aij in the ith row and j th column. The cofactor corresponding to the entry aij is
Cij = (-1)i+j Mij

In details:
a11 . . . i+j ai-1,1 = (-1) ai+1,1 . . . an1 ··· ··· ··· ··· a1,j-1
. . .

a1,j+1 ai-1,j+1 ai+1,j+1 an,j+1

··· ··· ··· ···

a1n
. . .

Cij

ai-1,j-1 ai+1,j-1
. . .

ai-1,n ai+1,n
. . .

an,j-1

ann



Theorem 9.8 (Determinant expansion by minors)
n

|A| =
k=1

aik Cik

Remark 9.5 In the previous theorem the expansion of the determinant was given according to the ith row. Similar expansion of the determinant is possible according to any column as well. Theorem 9.9 If the entries of the ith row are multiplied by the cofactors corresponding to a dierent j th row, then 0 is obtained:
n

aik Cjk = 0..
k=1
Proof:

j th rows are the same, therefore its determinant is zero. A can be given in the form:

This is the expansion of the determinant of a matrix in which the ith and

From the previous two theorems immediately follows that the inverse of a matrix
A-1 =
C11 |A| C12 |A| C21 |A| C22 |A|

C1n |A|

. . .

C2n |A|

. . .

... ...

..

Cn1 |A| Cn2 |A|

.

.

...

Cnn |A|

. . .

Finding determinants using elementary basis transformations
Previously we found that similar matrices have the same determinant and the determinant of a product matrix is the product of their determinants. Let us assume that the matrix A has the partitioned form:
A= A11 A21 A12 A22

and B =

A11 A21

0 E

It can be easily checked that
B-1 = A-1 11 -A21 A-1 11 0 E ,

6

9.3.

DETERMINANT EXPANSION BY MINORS

therefore
B-1 AB = A-1 11 -A21 A-1 11 0 E A11 A21 A12 A22 · A11 A21 A11 A21 0 E 0 E =

=

E A-1 A12 11 0 A22 - A21 A-1 A12 11

is similar to the matrix A and thus
|A| = |A11 | · |A22 - A21 A-1 A12 |. 11

We have to draw the attention to the fact that the A22 - A21 A-1 A12 matrix 11 block is obtained if we perform basis transformation choosing the A11 to be the pivoting block. It follows that it can be proved by mathematical induction that the determinant of a matrix A can be obtained if we multiply the pivot elements of the basis transformations exchanging the original basis to the columns of the matrix A and it is multiplied by sgn(), where is the permutation mapping the row indices to the column indices of the pivot element.

Example 9.4 Calculate the determinant of the matrix:
1 1 A= 2 1
Solution:



0 1 1 3 0 1 1 1

1 1 3 2

0 1 1 1 0 1 0 2 1 0 1 1 0 1 0 2

1 1 2 1

0 1 0 1 0 1 0 1 1 1 1 1 1 3 1 3 1 3 2 1



The product of pivot elements is 2, the corresponding permutation is
= 1 1 2 4 3 3 4 2

and sgn() = -1, because the number of inversions is 3, therefore the determinant of the given matrix is |A| = -2.

Exercise 9.1 Find the determinant of the subsequent matrices of order 3:


a) d) g)

Determine the determinant of the following matrices of order 4:


1 4 -3 -1 -4 -2 e) 0 -1 1 0 -4 3 0 -1 -2 h) 1 1 2

1 0 0

-5 -2 5 -3 -1 0



b)

-5 -2 2 3 1 -3 0 0 -4 -5 4 0 3 -2 2 -3 3 3 1 3 -3 -5 5 5 1 5 -4 -3 5 -2 -5

c) f)

1 1 2 -4 4 -4 -4 4 -4 5 0 5 -4 1 -1 3 -2 -3



a)

-1 -2 -2 4 -4 -4 -5 -4 -4 -5 3 -1 1 1 4 2

b)

-2 4 1 -2 2 1 0 3 5 3 -3 -1

7

9.3.

DETERMINANT EXPANSION BY MINORS

-4 -1 5 -1 1 1 2 -1 5 4 c) -5 0 -4 -5 d) -2 -4 -5 2 5 0 1 -5 1 3 5 4 0 4 4 -2 3 5 4 3 4 0 -4 -3 -2 2 1 1 0 e) 4 3 -1 5 f) -3 -5 0 -1 5 2 3 4 3 -1 -1 4 0 -1 0 0 1 -3 5 -5 5 0 -2 1 -5 0 3 3 4 0 g -4 -2 -2 -5 h) -1 4 -1 -4 1 4 3 2 0 0 -2 5 -1 -5 3 3 -3 5 -3 4 3 -2 1 5 0 2 i) 2 -2 -1 -2 j) 0 -4 0 -1 1 4 4 5 1 -1 3 3 3 -3 -5 2 -2 1 0 2 -4 -5 -3 1 5 5 4 k) 5 4 -4 -2 l) -1 -2 -1 2 0 5 -3 2 4 5 0 -3 5 -4 -5 0 -4 -4 3 -3 0 2 4 -4 2 -3 -1 0 3 -1 m) -4 1 3 1 n) -2 5 5 -2 -4 1 0 0 0 0 1 4 4 4 2 -3 2 1 2 0 3 2 -1 1 o) -4 -3 5 -3 3 2 5 -3



8



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