資訊科學數學 14 : determinants & inverses

11

資訊科學數學資訊科學數學 14 14 ::

Determinants & InversesDeterminants & Inverses

陳光琦助理教授陳光琦助理教授 (Kuang-Chi Chen)(Kuang-Chi Chen)[email protected]@mail.tcu.edu.tw

22

Linear Equations and MatricesLinear Equations and Matrices

DeterminantsDeterminants

33

3.1 Determinants3.1 Determinants• With each With each nnnn matrix matrix AA it is possible to associate a it is possible to associate a

scalar scalar det(det(AA)), called the , called the determinantdeterminant of the matrix, w of the matrix, whose value will tell us whether the matrix is hose value will tell us whether the matrix is singular singular or notor not..

• Case 1: 1Case 1: 11 matrices 1 matrices

- If - If AA = ( = (aa), then ), then AA will have a multiplicative inverse will have a multiplicative inverse iff iff aa≠0 ≠0 ..

- - AA is nonsingular iff det( is nonsingular iff det(AA))≠≠0 .0 .

44

222 Matrices2 Matrices

• Case 2: 2Case 2: 22 matrices 2 matrices

- Let - Let AA = . = .

- - AA will be will be nonsingularnonsingular iff iff det(det(AA) = ) = aa1111aa2222 – – aa1212aa2121≠ ≠ 00 . .

11 12

21 22

a a

a a

55

333 Matrices3 Matrices• Case 3: 3Case 3: 33 matrices 3 matrices

- Let - Let AA = . = .

- - AA will be will be nonsingularnonsingular iff iff

det(det(AA) = ) = aa1111aa2222aa3333 + + aa1212aa3131aa23 23 + + aa1313aa2121aa3232 – – aa1111aa3232aa23 23 – – aa1212

aa2121aa33 33 – – aa1313aa3131aa22 22 ≠ ≠ 00 . .

11 12 13

21 22 23

31 32 33

a a a

a a a

a a a

66

Example 4 & 5Example 4 & 5

• Example 4Example 4

If If AA = [ = [aa1111] is a 1] is a 11 matrix, then det(1 matrix, then det(AA) = ) = aa1111 . .


IfIf

⇒⇒ det(det(AA) = ) = aa1111aa2222 – – aa1212aa2121

⇒⇒ det(det(AA) = (2)(5) – (-3)(4)) = (2)(5) – (-3)(4) = 22 = 22

11 12

21 22

a aA

a a

54

32A

77

Example 6 & 7Example 6 & 7• Example 6Example 6

IfIf

⇒⇒ det(det(AA) =) = aa1111aa2222aa3333 + + aa1212aa3131aa23 23 + + aa1313aa2121aa3232

– – aa1111aa3232aa23 23 – – aa1212aa2121aa33 33 – – aa1313aa3131aa22 22


IfIf ⇒ ⇒ det(det(AA) = (1)(1)(2) + (3)(2)(1)) = (1)(1)(2) + (3)(2)(1) + (2)(3)(3)+ (2)(3)(3)

– – (3)(1)(3)(3)(1)(3) – (1)(1)(3)– (1)(1)(3) – (2)(2)(2) = 6– (2)(2)(2) = 6

333231

232221

131211

aaa

aaa

aaa

A

213

312

321

A

88

Properties of DeterminantsProperties of Determinants

• Theorem 3.1Theorem 3.1

The determinants of a matrix and its The determinants of a matrix and its transposetranspose are are eqequalual, i.e., det(, i.e., det(AA) = det() = det(AATT).).

99

Example 8Example 8• Example 8Example 8

IfIf

⇒ ⇒ det(det(AATT) = (1)(1)(2) + (3)(1)(2)) = (1)(1)(2) + (3)(1)(2) + (2)(3)(3)+ (2)(3)(3)

– – (3)(1)(3)(3)(1)(3) – (1)(1)(3)– (1)(1)(3) – (2)(2)(2)– (2)(2)(2)

= 6 = det(= 6 = det(AA))

233

112

321TA

213

312

321

A

1010

Theorem 3.2 & 3.3Theorem 3.2 & 3.3


If matrix If matrix BB results from matrix results from matrix AA by by interchanginginterchanging t two rows (or two columns) of wo rows (or two columns) of AA, then, then

det(det(BB) = -det() = -det(AA).).


If If two rows (or columns) of two rows (or columns) of AA are equal are equal, then, then

det(det(AA) = 0.) = 0.

1111

Example 9 & 10Example 9 & 10• Example 9Example 9

IfIf


IfIf

712

23 and 7

23

12

0

321

701

321

1212

Theorem 3.4Theorem 3.4


If a row (or column) of If a row (or column) of AA consists entirely of consists entirely of zeroszeros, t, then det(hen det(AA) = 0.) = 0.


0

000

654

321

1313

Theorem 3.5Theorem 3.5• Theorem 3.5Theorem 3.5

If If BB is obtained from is obtained from AA by by multiplyingmultiplying a row (colum a row (column) of n) of AA by a real number by a real number cc, then, then

det(det(BB) = ) = cc det( det(AA)) . .


1814641

1132

121

312

121

62

1414

Example 13Example 13


0032

141

151

121

32

341

351

321

2

682

351

321

1515


If If BB = [ = [bbijij] is obtained from ] is obtained from AA = [ = [aaijij] by adding to eac] by adding to eac

h element of the h element of the rrthth row (column) of row (column) of AA a constant a constant cc ti times the corresponding element of the mes the corresponding element of the ssthth row (colum row (column) n) rr≠≠ss of of AA, then det(, then det(BB) = det() = det(AA) .) .


101

312

905

101

312

321

1616


If a matrix If a matrix AA = [ = [aaijij] is ] is upper (lower) triangularupper (lower) triangular, then, , then,

then det(then det(AA) = ) = aa11 11 aa22 22 … … aannnn . .

• Corollary 1.3Corollary 1.3

The determinant of a The determinant of a diagonal matrixdiagonal matrix is the product o is the product of the entries on its main diagonal.f the entries on its main diagonal.

1717

Example 15Example 15


2 3 4 3 0 0 5 0 0

0 4 5 , 2 5 0 , 0 4 0

0 0 3 6 8 4 0 0 6

A B C

120)det( ,60)det( ,24)det( CBA

1818

Elementary OperationsElementary Operations• Elementary row and elementary column operationsElementary row and elementary column operations

I - Interchange rows (columns) I - Interchange rows (columns) ii and and jj : :

rrii ⇔ ⇔ rrjj ( (ccii ⇔ ⇔ ccjj ) )

II - Replace row (column) II - Replace row (column) ii by a nonzero value by a nonzero value kk times times row (column) row (column) ii : :

krkrii ⇔ ⇔ rrii ( (kckcii ⇔ ⇔ ccii ) )

III - Replace row (column) III - Replace row (column) jj by a nonzero value by a nonzero value kk times times row (column) row (column) i+ i+ row (column) row (column) jj : :

krkrii + + rrjj ⇔ ⇔ rrjj ( (kckcii + + ccjj ⇔ ⇔ ccjj ) )

1919

… … then …then …

det( ) det( ),

det( ) det( )

det ( ) det( ),

i j

i i

i j j

r r

kr r

kr r r

A A i j

A k A

A A i j

2020

Example 16Example 16• E.g. 16E.g. 16

642

523

234

A

13 32

1 3

4 3 2

det( ) 2det( ) 2det ( 3 2 5 )

1 2 3

4 3 2 1 2 3

2det( 3 2 5 ) ( 1) 2det( 3 2 5 )

1 2 3 4 3 2

r r

r r

A A

2121

Example 16Example 16 (cont’d)(cont’d)

1 2 2

1 3 3

52 3 38

-3r4

304

1 2 3 1 2 3

2det( 3 2 5 ) 2det( 0 -8 4 )

4 3 2 0 -5 10

1 2 3 1 2 3

2det( 0 -8 4 ) 2det( 0 -8 4 )

0 -5 10 0 0

r rr r r

r r r

det( ) 2(1)( 8)( 30 / 4) 120A

2222


The determinant of a product of two matrices is the prThe determinant of a product of two matrices is the product of their determinants oduct of their determinants det(det(ABAB) = det() = det(AA)det()det(BB)) . .


43

21A

21

12B

2A 5B

510

34AB BAAB 10

2323

Example 17 Example 17 (cont’d)(cont’d)

• RemarkRemark

ABAB≠≠BABA

||BABA| = || = |BB| || |AA|= -10 = ||= -10 = |ABAB||

107

01BA

2424

Corollary 3.2Corollary 3.2


If If AA is is nonsingularnonsingular, then , then det(det(AA) ) ≠≠ 0 0,,

thus thus det(det(AA-1-1) = 1/det() = 1/det(AA))..

If If AA is singular, then det( is singular, then det(AA) = 0) = 0

( 1 = |( 1 = |II| = || = |AAAA-1-1| = || = |AA| || |AA-1-1| )| )

2525


43

21A 2)det( A

1 2 1

3/ 2 1/ 2A

)det(

1

2

1)det( 1

AA

2626

Cofactor Expression and Cofactor Expression and ApplicationsApplications

2727

3.2 Cofactor Expression and 3.2 Cofactor Expression and ApplicationsApplications

Cofactor expression and applicationsCofactor expression and applications

• Definition – Minor and cofactorDefinition – Minor and cofactor

Let Let AA = [ = [aaijij] be an ] be an nnnn matrix. Let matrix. Let MMijij be the ( be the (nn-1)-1)

((nn-1) -1) submatrixsubmatrix of of AA obtained by deleting the obtained by deleting the iithth row row and and jjthth column of column of AA. The . The determinant det(determinant det(MMijij)) is calle is calle

d the d the minorminor of of aaijij. The . The cofactor cofactor AAijij of of aaijij is defined as is defined as

)( det)1( ijji

ij MA

2828


LetLet

217

654

213

A

12

23

31

4 6det( ) 8 42 34

7 2

3 1det( ) 3 7 10

7 1

1 2det( ) 6 10 16

5 6

M

M

M

1 212 12

2 323 23

3 131 31

( 1) det( ) ( 1)( 34) 34

( 1) det( ) ( 1)(10) 10

( 1) det( ) (1)( 16) 16

A M

A M

A M

2929



Let Let AA = [ = [aaijij] be an ] be an nnnn matrix. Then matrix. Then

for each 1for each 1≤ ≤ ii ≤ ≤ nn,,

det(det(AA) = ) = aaii11AAii11 + + aaii22AAii22 + … + + … + aaininAAinin , and , and

for each 1for each 1≤ ≤ jj ≤ ≤ nn,,

det(det(AA) = ) = aa11jjAA11jj + + aa22jjAA22jj + … + + … + aanjnjAAnjnj . .

3030

Example 2Example 2To evaluate the determinantTo evaluate the determinant

3202

3003

3124

4321

3 1 3 2

3 3 3 4

1 2 3 42 3 4 1 3 4

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

3 0 0 30 2 3 2 2 3

2 0 2 3

1 2 4 1 2 3

( 1) (0) -4 2 3 ( 1) ( 3) 4 2 1

2 0 3 2 0 2

( 1)(3)(20) 0 0 ( 1)( 3)( 4) 48

3131

Example 3Example 3Consider the determinant of the matrixConsider the determinant of the matrix

4 1 4

1 2 1

3 1 4

4

1 2 3 4 1 2 3 5

4 2 1 3 4 2 1 1

3 0 0 3 3 0 0 0

2 0 2 3 2 0 2 5

2 3 5 0 4 6

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

0 2 5 0 2 5

( 1) (3)( 2)( 8) 48

c c c

r r r

3232


If If AA = [ = [aaijij] be an ] be an nnnn matrix, then matrix, then

aaii11AAkk11 + + aaii22AAkk22 + … + + … + aaininAAknkn = 0, for = 0, for ii≠≠kk , ,

aa11jjAA11kk + + aa22jjAA22kk + … + + … + aanjnjAAnknk = 0, for = 0, for jj≠≠kk . .

3333


033142191

032145194

354

211

1424

311

1925

321

254

132

321

231322122111

233322322131

3223

2222

1221

AaAaAa

AaAaAa

A

A

A

A

3434

AdjointAdjoint• Definition – AdjointDefinition – Adjoint

Let Let AA = [ = [aaijij] be an ] be an nnnn matrix. The matrix. The nnnn matrix matrix adjadj AA,,

called the called the adjoint of adjoint of AA, is the matrix whose , is the matrix whose jj, , iithth ele element is the ment is the cofactor cofactor AAijij of of aaijij . Thus . Thus

nnnn

n

n

AAA

AAA

AAA

adjA

21

22212

12111

3535

RemarkRemark

• RemarkRemark

The adjoint of The adjoint of AA is formed by taking the is formed by taking the transptransposeose of the matrix of of the matrix of cofactorscofactors AAijij of the elemen of the elemen

ts of ts of AA..

3636


Compute adj Compute adj AA

301

265

123

A

3737

SolutionSolution

1 1

11

1 2

12

1 3

13

2 1

21

2 2

22

2 3

23

6 21 18

0 3

5 21 17

1 3

5 61 6

1 0

2 11 6

0 3

3 11 10

1 3

3 21 2

1 0

A

A

A

A

A

A

2865

231

125

131

1026

121

3333

2332

1331

A

A

A

2826

11017

10618

adj Then, A

3838



If If AA = [ = [aaijij] be an ] be an nnnn matrix, then matrix, then

AA(adj (adj AA) = (adj ) = (adj AA))AA = det( = det(AA) ) IInn . .

3939

Example 6Example 6• E.g. 6 E.g. 6

Consider the matrixConsider the matrix

3 2 1 18 6 10 94 0 0 1 0 0

5 6 2 17 10 1 0 94 0 94 0 1 0

1 0 3 6 2 28 0 0 94 0 0 1

and

18 6 10 3 2 1 1 0 0

17 10 1 5 6 2 94 0 1 0

6 2 28 1 0 3 0 0 1

301

265

123

A

4040

Corollary 3.3Corollary 3.3• Corollary 3.3Corollary 3.3

If If AA = [ = [aaijij] be an ] be an nnnn matrix and det( matrix and det(AA))≠≠0, then0, then

A

A

A

A

A

A

A

A

A

A

A

AA

A

A

A

A

A

adjAA

A

detdetdet

detdetdet

detdetdet

det

1 1

4141


Consider the matrixConsider the matrix

Then det(Then det(AA) = -94, and) = -94, and

3 2 1

5 6 2

1 0 3

A

18 6 1094 94 94

1 17 10 194 94 94

6 128294 94 94

1

det( )A adjA

A

4242



A matrix A matrix AA = [ = [aaijij] is ] is nonsingularnonsingular iff iff det(det(AA) ) ≠≠ 0 0..


For an For an nnnn matrix matrix AA, the homogeneous system , the homogeneous system AAx x = 0 has a = 0 has a nontrival solutionnontrival solution iff iff det(det(AA) = 0) = 0..

4343


Let Let AA be a 4x4 matrix with det( be a 4x4 matrix with det(AA) = -2) = -2

(a) describe the set of all solutions to the homoge(a) describe the set of all solutions to the homogeneous system neous system AAxx = 0. = 0.

(b) If (b) If AA is transformed to reduced row echelon form is transformed to reduced row echelon form BB, , what is what is BB??

(c) Given an expression for a solution to the linear syst(c) Given an expression for a solution to the linear system em AAxx = = bb, where , where bb = [ = [bb11 , , bb22 , , bb33 , , bb44 ] ]TT . .

(d) Can the linear system (d) Can the linear system AAxx = = bb have more than one sol have more than one solution? Explain.ution? Explain.

(e) Does (e) Does AA-1-1 exist? exist?

4444

Solutions of Example 8Solutions of Example 8• SolutionsSolutions

(a) Since det((a) Since det(AA))≠≠0, 0, AxAx = 0 has = 0 has only the trivial solutiononly the trivial solution..

(b) Since det((b) Since det(AA))≠≠0, 0, AA is a nonsingular matrix, so is a nonsingular matrix, so BB = = IInn

(c) A solution to the given system is given by (c) A solution to the given system is given by xx = = AA-1-1bb

(d) No. The solution is unique.(d) No. The solution is unique.

(e) Yes. (e) Yes.

4545

Nonsingular EquivalenceNonsingular Equivalence• List of nonsingular equivalenceList of nonsingular equivalence

The following statements are equivalent.The following statements are equivalent.

1.1. AA is is nonsingularnonsingular..

2.2. xx = 0 = 0 is the is the only solutiononly solution to to AxAx = 0. = 0.

3.3. AA is is row equivalencerow equivalence to to IInn . .

4. The linear system 4. The linear system AAxx = = bb has a has a unique solutionunique solution for for every every nn1 matrix 1 matrix bb..

5. 5. det(det(AA))≠≠00 . .

4646


• Linearly independentLinearly independent

• NonsingularNonsingular

• Trivial solution Trivial solution xx = 0 to = 0 to AxAx = 0 = 0

• det(det(AA) ) ≠≠ 0 0

4747


• Linearly dependentLinearly dependent

• SingularSingular

• Nontrivial solution to Nontrivial solution to AAxx = 0 = 0

• det(det(AA) = 0) = 0

4848

Cramer’s RuleCramer’s RuleTheorem 3.13 (Cramer’s Rule)Theorem 3.13 (Cramer’s Rule)

Let Let aa1111xx11 + + aa1212xx22 + … + + … + aa11nnxxnn = = bb11

aa2121xx11 + + aa2222xx22 + … + + … + aa22nnxxnn = = bb22

……

aann11xx11 + + aann22xx22 + … + + … + aannnnxxnn = = bbnn

Then,Then,

xx11 = det( = det(AA11)/det()/det(AA)) , , xx22 = det( = det(AA22)/det()/det(AA)) , … , , … ,

xxnn = det( = det(AAnn)/det()/det(AA)) . .

4949

Cramer’s RuleCramer’s Rule

Cramer’s Rule for solving the linear system Cramer’s Rule for solving the linear system AAxx = = bb, wh, where ere AA is is nnnn, is as follows:, is as follows:

Step 1. Compute det(Step 1. Compute det(AA). If det(). If det(AA) = 0, Cramer’s rule is ) = 0, Cramer’s rule is not applicable. Use not applicable. Use Gauss-Jordan ReductionGauss-Jordan Reduction..

Step 2. If det(Step 2. If det(AA))≠≠0, for each 0, for each ii,,

xxii = det( = det(AAii)/det()/det(AA)) , ,

where where AAii is the matrix obtained from is the matrix obtained from AA by replacing t by replacing t

he he iithth column of column of AA by by bb. .

5050

Example 9Example 9

• Consider the following linear system:Consider the following linear system:

-2-2xx11 + 3 + 3xx22 – – xx33 = 1 = 1

xx11 + 2 + 2xx22 – – xx33 = 4 = 4

-2-2xx11 – 2 – 2xx22 + + xx33 = -3 = -3

ThenThen 2 3 1

1 2 1 2

2 1 1

A

5151

Example 9Example 9 (cont’d) (cont’d)

2

2 1 1

1 4 1

2 3 1 63

2x

A

1

1 3 1

4 2 1

3 1 1 42

2x

A

3

2 3 1

1 2 4

2 1 3 84

2x

A

Hence,Hence,

5252

Polynomial Interpolation Polynomial Interpolation RevisitedRevisited

• Polynomial Interpolation RevisitedPolynomial Interpolation Revisited

To find a quadratic polynomial that interpolates the To find a quadratic polynomial that interpolates the following points:following points:

((xx11, , yy11), (), (xx22, , yy22), (), (xx33, , yy33),),

where where xx11≠≠xx22 , , xx11≠ ≠ xx33 , , xx22≠ ≠ xx33 . .

5353

… … more …more …

The polynomial has the form:The polynomial has the form:

yy = = aa22xx22 + + aa11xx + + aa00 . .

The corresponding linear systemThe corresponding linear system

yy11 = = aa22xx1122 + + aa11xx11 + + aa00 , ,

yy22 = = aa22xx2222 + + aa11xx22 + + aa00 , ,

yy33 = = aa22xx3322 + + aa11xx33 + + aa00 . .

5454

… … more …more …

The coefficient matrixThe coefficient matrix

The Vandermount determinantThe Vandermount determinant

((xx11 – – xx22 )( )( xx11 – – xx33 )( )( xx22 – – xx33 ) )

1

1

1

3

2

3

2

2

2

1

2

1

xx

xx

xx

5656

Linear Equations and MatricesLinear Equations and Matrices

LU-FactorizationLU-Factorization

5757


• 1.8 LU-Factorization1.8 LU-Factorization

If a square matrix can be reduced to upper If a square matrix can be reduced to upper triangular form using only 3 row operations, triangular form using only 3 row operations, then it is possible to represent the then it is possible to represent the reduction reduction processprocess in terms of a matrix factorization. in terms of a matrix factorization.

5858

Type I OperationType I Operation

An elementary matrix of type I is a matrix obtained by An elementary matrix of type I is a matrix obtained by interchanginginterchanging two rows of identity matrix two rows of identity matrix II..

ExampleExample1

1 1

1

1

0 1 0 0 1 0

1 0 0 , 1 0 0

0 0 1 0 0 1

Interchange the first row and

the second row of

Interchange the first column

and the second column of

E E

E A

A

AE

A

5959

Type II OperationType II Operation

An elementary matrix of type II is a matrix obtained An elementary matrix of type II is a matrix obtained by by multiplyingmultiplying a row of a row of II byby a nonzero constant.a nonzero constant.

ExampleExample

12 2

2

2

1 0 0 1 0 0

0 1 0 , 0 1 0

0 0 3 0 0 1/ 3

Multiply the third row of by 3

Multiply the third column of by 3

E E

E A A

AE A

6060

Type III OperationType III Operation

An elementary matrix of type III is a matrix obtained An elementary matrix of type III is a matrix obtained from from II by adding a multiple of one row to another row. by adding a multiple of one row to another row.

ExampleExample1

3 3

3

3

1 0 3 1 0 3

0 1 0 , 0 1 0

0 0 1 0 0 1

Add three times the third row to

the first row of

Add three times the third column

to the first column of

E E

E A

A

AE

A

6161

In GeneralIn GeneralIn general, the elementary matrix by adding In general, the elementary matrix by adding mm times times of row of row ii to row to row jj

1000

10

10

1

,

1000

10

10

1

133

m

E

m

E

Row j

Column iaji

6262

TheoremTheorem• TheoremTheorem

If If AA and and BB are are nonsingular squarenonsingular square matrices, then matrices, then ABAB i is also s also nonsingularnonsingular..

i.e. (i.e. (ABAB))-1-1 = = BB-1-1AA-1-1 . .

In general, if In general, if EE11, , EE22, …, , …, EEkk are all are all nonsingularnonsingular, then t, then t

he product he product EE11EE22 … … EEkk is also is also nonsingularnonsingular and and

((EE11EE22 … … EEkk ) )-1-1 = = EEkk-1-1 … … EE22

-1 -1 EE11-1-1 . .

6363

Row EquivalenceRow Equivalence

• A matrix A matrix BB is is row equivalent torow equivalent to AA if there exists a fin if there exists a finite sequence ite sequence EE11EE22 … … EEkk of elementary matrices such of elementary matrices such

that that BB = = EEkk … … EE22 EE11AA . .

6464


• LU-FactorizationLU-Factorization

If a square matrix can be reduced to upper If a square matrix can be reduced to upper triangular form using only 3 row operations, triangular form using only 3 row operations, then it is possible to represent the then it is possible to represent the reduction reduction processprocess in terms of a matrix factorization. in terms of a matrix factorization.

6565

ExampleExample• LetLet

914

251

242

A

1

1 21

2 4 2 2 4 2

1 5 2 0 3 1

4 1 9 4 1 9

1 0 0

1/ 2 1 0 , 1/ 2

0 0 1

E A

E a

6666

(cont’d)(cont’d)

2

2 31

2 4 2 2 4 2

0 3 1 0 3 1

4 1 9 0 9 5

1 0 0

0 1 0 , 2

2 0 1

E A

E a

3

3 32

2 4 2 2 4 2

0 3 1 0 3 1

0 9 5 0 0 8

1 0 0

0 1 0 , 3

0 3 1

E A

E a

6767

(cont’d)(cont’d)

3 2 1

1 1 1 13 2 1 1 2 3

1 1 11 2 3

2 4 2

0 3 1

0 0 8

( )

1 0 01 0 0 1 0 0

11 0 0 1 0 0 1 0

22 0 1 0 3 10 0 1

E E E A U

A E E E U E E E U LU

L E E E

資訊科學數學 14 : determinants & inverses

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