Elementary Linear Algebra Anton & Rorres, 9 th Edition

Elementary Linear AlgebraAnton & Rorres, 9th Edition

Lecture Set – 02Chapter 2: Determinants

112/04/20 Elementary Linear Algebra 2

Chapter Content

Determinants by Cofactor Expansion Evaluating Determinants by Row Reduction Properties of the Determinant Function A Combinatorial Approach to Determinants


2-1 Minor and Cofactor

Definition Let A be mn

The (i,j)-minor of A, denoted Mij is the determinant of the (n-1) (n-1) matrix formed by deleting the ith row and jth column from A

The (i,j)-cofactor of A, denoted Cij, is (-1)i+j Mij

Remark Note that Cij = Mij and the signs (-1)i+j in the definition of

cofactor form a checkerboard pattern:

...

...

...

...


2-1 Example 1

Let

The minor of entry a11 is

The cofactor of a11 is C11 = (-1)1+1M11 = M11 = 16

Similarly, the minor of entry a32 is

The cofactor of a32 is C32 = (-1)3+2M32 = -M32 = -26

8 4 1

6 5 2

4 1 3

A

168 4

6 5

8 4 1

6 5 2

4 1 3

11

M

266 2

4 3

8 4 1

6 5 2

4 1 3

32

M


2-1 Cofactor Expansion The definition of a 3×3 determinant in terms of minors and

cofactors det(A) = a11M11 +a12(-M12)+a13M13

= a11C11 +a12C12+a13C13

this method is called cofactor expansion along the first row of A

Example 2

10)11)(1()4(3 4 5

4- 20

2 5

3 2 1

2 4

3 43

2 4 5

3 4 2

0 1 3

)det(

A

2-1 Cofactor Expansion det(A) =a11C11 +a12C12+a13C13 = a11C11 +a21C21+a31C31

=a21C21 +a22C22+a23C23 = a12C12 +a22C22+a32C32

=a31C31 +a32C32+a33C33 = a13C13 +a23C23+a33C33

Theorem 2.1.1 (Expansions by Cofactors) The determinant of an nn matrix A can be computed by multiplying

the entries in any row (or column) by their cofactors and adding the resulting products; that is, for each 1 i, j n

det(A) = a1jC1j + a2jC2j +… + anjCnj

(cofactor expansion along the jth column)

and

det(A) = ai1Ci1 + ai2Ci2 +… + ainCin

(cofactor expansion along the ith row)


2-1 Example 3 & 4 Example 3

cofactor expansion along the first column of A

Example 4 smart choice of row or column

det(A) = ?


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

0 1 5

2 4

0 1 )2(

2 4

3 43

2 4 5

3 4 2

0 1 3

)det(

A

1002

12-01

2213

1-001

A


2-1 Adjoint of a Matrix If A is any nn matrix and Cij is the cofactor of aij, then the

matrix

is called the matrix of cofactors from A. The transpose of this matrix is called the adjoint of A and is denoted by adj(A)

Remarks If one multiplies the entries in any row by the corresponding

cofactors from a different row, the sum of these products is always zero.

nnnn

n

n

C CC

C CC

C CC

21

22221

11211

2-1 Example 5 Let

a11C31 + a12C32 + a13C33 = ?

Let


333231

232221

131211

aaa

aaa

aaa

A

131211

232221

131211

'

aaa

aaa

aaa

A


2-1 Example 6 & 7

Let

The cofactors of A are: C11 = 12, C12 = 6, C13 = -16, C21 = 4, C22 = 2, C23 = 16, C31 = 12, C32 = -10, C33 = 16

The matrix of cofactor and adjoint of A are

The inverse (see below) is

042

361

123

A

16 10 12

16 2 4

16 6 12

16 16 16

10 2 6

12 4 12

)(adj A

16 16 16

10 2 6

12 4 12

64

1)(adj

)det(

11 AA

A


Theorem 2.1.2 (Inverse of a Matrix using its Adjoint) If A is an invertible matrix, then

Show first that )(adj

)det(

11 AA

A

)det(2211 ACaCaCa ininiiii

IAAA )det()(adj

)(02211 jiCaCaCa jninjiji

Theorem 2.1.3 If A is an n × n triangular matrix (upper triangular, lower

triangular, or diagonal), then det(A) is the product of the entries on the main diagonal of the matrix; det(A) = a11a22…ann

E.g.


44434241

333231

2221

11

0

00

000

aaaa

aaa

aa

a

A

?

40000

89000

67600

15730

38372

2-1 Prove Theorem 1.7.1c A triangular matrix is invertible if and only if its diagonal

entries are all nonzero


2-1 Prove Theorem 1.7.1d The inverse of an invertible lower triangular matrix is

lower triangular, and the inverse of an invertible upper triangular matrix is upper triangular


Theorem 2.1.4 (Cramer’s Rule) If Ax = b is a system of n linear equations in n unknowns such

that det(I – A) 0 , then the system has a unique solution. This solution is

where Aj is the matrix obtained by replacing the entries in the column of A by the entries in the matrix b = [b1 b2 ··· bn]T


)det(

)det(,,

)det(

)det(,

)det(

)det( 22

11 A

Ax

A

Ax

A

Ax n

n


2-1 Example 9 Use Cramer’s rule to solve

Since

Thus,

832

30643

62

321

321

31

xxx

xxx

x x

8 2 1

30 4 3

6 0 1

,

3 8 1

6 30 3

2 6 1

,

3 2 8

6 4 30

2 0 6

,

3 2 1

6 4 3

2 0 1

321 AAAA

11

38

44

152

)det(

)det(,

11

18

44

72

)det(

)det(,

11

10

44

40

)det(

)det( 33

22

11

A

Ax

A

Ax

A

Ax


Chapter Content



Theorems Theorem 2.2.1

Let A be a square matrix If A has a row of zeros or a column of zeros, then det(A)

= 0.

Theorem 2.2.2 Let A be a square matrix det(A) = det(AT)


Theorem 2.2.3 (Elementary Row Operations) Let A be an nn matrix

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

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

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

Example 1

333231

232221

131211

k k k

)det(

aaa

aaa

aaa

B

2-2 Example of Theorem 2.2.3



Theorem 2.2.4 (Elementary Matrices) Let E be an nn elementary matrix

If E results from multiplying a row of In by k, then det(E) = k

If E results from interchanging two rows of In, then det(E) = -1

If E results from adding a multiple of one row of In to another, then det(E) = 1

Example 2

Theorem 2.2.5 (Matrices with Proportional Rows or Columns) If A is a square matrix with two proportional rows or two

proportional column, then det(A) = 0

Example 3



2-2 Example 4 (Using Row Reduction to Evaluate a Determinant) Evaluate det(A) where

Solution:

162

9 63

510

-

A

.

1 6 2

5 1 0

3 2 1

3

1 6 2

5 1 0

9 6 3

1 6 2

9 6 3

5 1 0

)det(

A

The first and second rows of A are interchanged.

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


2-2 Example 4 (continue)

.

165)1)(55)(3(

1 0 0

5 1 0

3 2- 1

)55)(3(

55- 0 0

5 1 0

3 2- 1

3

5- 10 0

5 1 0

3 2- 1

3

-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.

1 6 2

5 1 0

3 2 1

3)det(

A

2-2 Example 5 Using column operation to evaluate a determinant

Compute the determinant of


5-137

0360

6072

3001

A

2-2 Example 6 Row operations and cofactor expansion

Compute the determinant of


3573

5142

11-21

62-53

A


Chapter Content



2-3 Basic Properties of Determinant Since a common factor of any row of a matrix can be

moved through the det sign, and since each of the n row in kA has a common factor of k, we obtain

det(kA) = kndet(A)

There is no simple relationship exists between det(A), det(B), and det(A+B) in general.

In particular, we emphasize that det(A+B) is usually not equal to det(A) + det(B).

2-3 Example 1

det(A+B) ≠det(A)+det(B)


83

34,

31

13,

52

21ABBA


Theorems 2.3.1 Let A, B, and C be nn matrices that differ only in a single

row, say the r-th, and assume that the r-th row of C can be obtained by adding corresponding entries in the r-th rows of A and B.

Then det(C) = det(A) + det(B)

The same result holds for columns. Let

2221

1211

aa

aaA

2221

1211

bb

aaB

2-3 Example 2

Using Theorem 2.3.1


110

3 0 2

5 7 1

det

741

3 0 2

5 7 1

det

)1(71401

3 0 2

5 7 1

det


Lemma 2.3.2 If B is an nn matrix and E is an nn elementary matrix,

then det(EB) = det(E) det(B)

Remark: If B is an nn matrix and E1, E2, …, Er, are nn elementary

matrices, then

det(E1 E2 · · · Er B) = det(E1) det(E2) · · · det(Er) det(B)

Theorem 2.3.3 (Determinant Test for Invertibility) A square matrix A is invertible if and only if det(A) 0

Example 3


642

101

321

A


Theorem 2.3.4

If A and B are square matrices of the same size, then det(AB) = det(A) det(B)

Example 4

143

172,

85

31,

12

13ABBA

Theorem 2.3.5

If A is invertible, then


)det(

1)det( 1

AA


2-4 Linear Systems of the Form Ax = x Many applications of linear algebra are concerned with

systems of n linear equations in n unknowns that are expressed in the form Ax = x, where is a scalar

Such systems are really homogeneous linear in disguise, since the expresses can be rewritten as (I – A)x = 0


2-3 Eigenvalue and Eigenvector The eigenvalues of an nn matrix A are the number for

which there is a nonzero x 0 with Ax = x.

The eigenvectors of A are the nonzero vectors x 0 for which there is a number with Ax = x.

If Ax = x for x 0, then x is an eigenvector associated with the eigenvalue , and vice versa.

2-3 Eigenvalue and Eigenvector Remark:

A primary problem of linear system (I – A)x = 0 is to determine those values of for which the system has a nontrivial solution.

Theorem (Eigenvalues and Singularity) is an eigenvalue of A if and only if I – A is singular,

which in turn holds if and only if the determinant of I – A equals zero:

det(I – A) = 0 (the so-called characteristic equation of A)



2-3 Example 5 & 6 The linear system

The characteristic equation of A is

The eigenvalues of A are = -2 and = 5 By definition, x is an eigenvector of A if and only if x is a nontrivial

solution of (I – A)x = 0, i.e.,

If = -2, x = [-t t]T – one eigenvector If = 5, x = [3t/4 t]T – the other eigenvector

221

121

24

3

xxx

xxx

2 4

3 1I

Axx A

0103or 0 2 4

3 1 )det( 2

AI

0

0

2 4

3 1

2

1

x

x


Theorem 2.3.6 (Equivalent Statements) If A is an nn matrix, then the following are

equivalent A is invertible. Ax = 0 has only the trivial solution The reduced row-echelon form of A as In

A is expressible as a product of elementary matrices Ax = b is consistent for every n1 matrix b Ax = b has exactly one solution for every n1 matrix b det(A) 0


2-4 Permutation A permutation of the set of integers {1,2,…,n} is an arrangement of

these integers in some order without omission repetition Example 1

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

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

1

2 3 4

3 4 2 4 2 3

4 3 4 2 3 2

2

1 3 4

3 4 1 4 1 3

4 3 4 1 3 1

3

1 2 4

2 4 1 4 1 2

4 2 4 1 2 1

4

1 2 3

2 3 1 3 1 2

3 2 3 1 2 1


2-4 Inversion An inversion is said to occur in a permutation (j1, j2, …,

jn) whenever a larger integer precedes a smaller one.

The total number of inversions occurring in a permutation can be obtained as follows:

Find the number of integers that are less than j1 and that follow j1 in the permutation;

Find the number of integers that are less than j2 and that follow j2 in the permutation;

Continue the process for j1, j2, …, jn. The sum of these number will be the total number of inversions in the permutation


2-4 Example 3

Determine the number of inversions in the following permutations: (6,1,3,4,5,2) (2,4,1,3) (1,2,3,4)

Solution: The number of inversions is 5 + 0 + 1 + 1 + 1 = 8 The number of inversions is 1 + 2 + 0 = 3 There no inversions in this permutation


2-4 Classifying Permutations A permutation is called even if the total number of inversions

is an even integer and is called odd if the total inversions is an odd integer

Example 4 The following table classifies the various permutations of {1,2,3}

as even or odd Permutation Number of Inversions

classification

(1,2,3)

(1,3,2)

(2,1,3)

(2,3,1)

(3,1,2)

(3,2,1)

0

1

1

2

2

3

even

odd

odd

even

even

odd


2-4 Elementary Product By an elementary product from an nn matrix A we shall

mean any product of n entries from A, no two of which come from the same row or same column.

Example The elementary product of the matrix

is

333231

232221

131211

aaa

aaa

aaa

312213312312322311

322113332112332211

aa aaa aaaa

aa aaa aaaa


2-4 Signed Elementary Product An nn matrix A has n! elementary products. There are

the products of the form a1j1a2j2 ··· anjn

, where (j1, j2, …, jn) is a permutation of the set {1, 2, …, n}.

By a signed elementary product from A we shall mean an elementary a1j1

a2j2 ··· anjn multiplied by +1 or -1.

We use + if (j1, j2, …, jn) is an even permutation and – if (j1, j2, …, jn) is an odd permutation


2-4 Example 6

List all signed elementary products from the matrices

2221

1211

aa

aa

333231

232221

131211

aaa

aaa

aaa

Elementary

Product

Associated

Permutation

Even or Odd Signed Elementary

Product

a11a22

a12a21

(1,2)

(2,1)

even

odd

a11a22

- a12a21

Elementary

Product

Associated

Permutation

Even or Odd Signed Elementary

Product

a11a22a33

a11a23a32

a12a21a33

a12a23a31

a13a21a32

a13a22a31

(1,2,3)

(1,3,2)

(2,1,3)

(2,3,1)

(3,1,2)

(3,2,1)

even

odd

odd

even

even

odd

a11a22a33

- a11a23a32

- a12a21a33

a12a23a31

a13a21a32

- a13a22a31


2-4 Determinant Let A be a square matrix. The determinant function is

denoted by det, and we define det(A) to be the sum of all signed elementary products from A. The number det(A) is called the determinant of A

Example 7

211222112221

1211det aaaa aa

aa

322311332112312213322113312312332211

333231

232221

131211

det aaaaaaaaaaaaaaaaaaa

aaa

aaa

aaa


2-4 Using mnemonic for Determinant The determinant is computed by summing the

products on the rightward arrows and subtracting the products on the leftward arrows

Remark: This method will not work for determinant of 44

matrices or higher!

2221

1211

aa

aa

333231

232221

131211

aaa

aaa

aaa

3231

2221

1211

aa

aa

aa

2-4 Example 8 Evaluate the determinants of


24

13A

987

654

321

B


2-4 Notation and Terminology We note that the symbol |A| is an alternative notation for det(A)

The determinant of A is often written symbolically asdet(A) = a1j1

a2j2 ··· anjn

where indicates that the terms are to be summed over all permutations (j1, j2, …, jn) and the + or – is selected in each term according to where the permutation is even or odd

Remark: 44 matrices need 4! = 24 signed elementary products 1010 determinant need 10! = 3628800 signed elementary products! Other methods are required.

Elementary Linear Algebra Anton & Rorres, 9 th Edition

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