aarjaneiro/planetmath huggingface dataset:
determinant inequalities

determinant inequalities

There are a number of interesting

inequalities

bounding the

determinant

of a

n

×

n

complex matrix

A

, where

ρ

is its

spectral radius

:

1)

|

det

⁡

(

A

)

|

≤

ρ

n

⁢

(

A

)

2)

|

det

⁡

(

A

)

|

≤

∏

i

=

1

n

(

∑

j

=

1

n

|

a

i

⁢

j

|

)

=

∏

i

=

1

n

∥

a

i

∥

1

3)

|

det

⁡

(

A

)

|

≤

∏

j

=

1

n

(

∑

i

=

1

n

|

a

i

⁢

j

|

)

=

∏

j

=

1

n

∥

a

j

∥

1

4)

|

det

⁡

(

A

)

|

≤

∏

i

=

1

n

(

∑

j

=

1

n

|

a

i

⁢

j

|

2

)

1

2

=

∏

i

=

1

n

∥

a

i

∥

2

5)

|

det

⁡

(

A

)

|

≤

∏

j

=

1

n

(

∑

i

=

1

n

|

a

i

⁢

j

|

2

)

1

2

=

∏

j

=

1

n

∥

a

j

∥

2

6) if

A

is

Hermitian

positive semidefinite

,

det

⁡

(

A

)

≤

∏

i

=

1

n

a

i

⁢

i

, with equality if and only if

A

is

diagonal

.

Inequalities 4)-6) are known as ”

Hadamard

’s inequalities”.

(Note that inequalities 2)-5) may suggest the idea that such inequalities could hold:

|

det

⁡

(

A

)

|

≤

∏

i

=

1

n

∥

a

i

∥

p

or

|

det

⁡

(

A

)

|

≤

∏

j

=

1

n

∥

a

j

∥

p

for any

p

∈

𝐍

; however, this is not true, as one can easily see with

A

=

[

1

1

-

1

1

]

and

p

=

3

. Actually, inequalities 2)-5) give the best possible estimate of this kind.)

Proofs:

1)

|

det

⁡

(

A

)

|

=

|

∏

i

=

1

n

λ

i

|

=

∏

i

=

1

n

|

λ

i

|

≤

∏

i

=

1

n

ρ

⁢

(

A

)

=

ρ

n

⁢

(

A

)

.

2) If

A

is

singular

, the thesis is trivial. Let then

det

⁡

(

A

)

≠

0

. Let’s define

B

=

D

⁢

A

,

D

=

d

⁢

i

⁢

a

⁢

g

⁢

(

d

11

,

d

22

,

⋯

,

d

n

⁢

n

)

,

d

i

⁢

i

=

(

∑

j

=

1

n

|

a

i

⁢

j

|

)

-

1

. (Note that

d

i

⁢

i

exist for any

i

, because

det

⁡

(

A

)

≠

0

implies no all-zero row exists.) So

∥

B

∥

∞

=

max

i

⁡

(

∑

j

=

1

n

|

b

i

⁢

j

|

)

=

1

and, since

ρ

⁢

(

B

)

≤

∥

B

∥

∞

, we have:

|

det

⁡

(

B

)

|

=

|

det

⁡

(

D

)

|

⁢

|

det

⁡

(

A

)

|

=

(

∏

i

=

1

n

∑

j

=

1

n

|

a

i

⁢

j

|

)

-

1

⁢

|

det

⁡

(

A

)

|

≤

ρ

n

⁢

(

B

)

≤

∥

B

∥

∞

n

=

1

,

from which:

|

det

⁡

(

A

)

|

≤

∏

i

=

1

n

(

∑

j

=

1

n

|

a

i

⁢

j

|

)

.

3) Same as 2), but applied to

A

T

.

4)-6) See related proofs attached to ”Hadamard’s inequalities”.

Title

determinant inequalities

Canonical name

DeterminantInequalities

Date of creation

2013-03-22 15:34:46

Last modified on

2013-03-22 15:34:46

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

12

Author

Andrea Ambrosio (7332)

Entry type

Result

Classification

msc 15A15

Generated on Fri Feb 9 18:24:00 2018 by

LaTeXML
determinants of some matrices of special form

determinants of some matrices of special form

Suppose

A

is

n

×

n

square matrix

,

u

,

v

are two column

n

-vectors, and

α

is a scalar. Then

det

⁡

(

A

+

u

⁢

v

T

)

=

det

⁡

A

+

v

T

⁢

adj

⁡

A

⁢

u

,

det

⁡

(

A

u

v

T

α

)

=

α

⁢

det

⁡

A

-

v

T

⁢

adj

⁡

A

⁢

u

,

where

adj

⁡

A

is the

adjugate

of

A

.

References

1

V.V. Prasolov,

Problems and Theorems in

Linear Algebra

,

American Mathematical Society

, 1994.

Title

determinants of some matrices of special form

Canonical name

DeterminantsOfSomeMatricesOfSpecialForm

Date of creation

2013-03-22 14:03:41

Last modified on

2013-03-22 14:03:41

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Result

Classification

msc 15A15

Related topic

BlockDeterminants

Generated on Fri Feb 9 18:24:02 2018 by

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Dieudonné theorem on linear preservers of the singular matrices

Dieudonné theorem on linear preservers of the singular matrices

Let

𝔽

be an arbitrary field. Consider

ℳ

n

⁢

(

𝔽

)

, the

vector space

of all

n

×

n

matrices over

𝔽

. Moreover, let

𝒢

⁢

ℒ

n

⁢

(

𝔽

)

be the full linear group of

nonsingular

n

×

n

matrices over

𝔽

.

Theorem 1

.

For a linear

automorphism

φ

:

M

n

⁢

(

F

)

⟶

M

n

⁢

(

F

)

the following conditions are

equivalent

:

(i)

∀

A

∈

M

n

⁢

(

F

)

:

det

⁡

(

A

)

=

0

⇒

det

⁡

(

φ

⁢

(

A

)

)

=

0

,

(ii)

either

∃

P

,

Q

∈

G

⁢

L

n

⁢

(

F

)

⁢

∀

A

∈

M

n

⁢

(

F

)

:

φ

⁢

(

A

)

=

P

⁢

A

⁢

Q

, or

∃

P

,

Q

∈

G

⁢

L

n

⁢

(

F

)

⁢

∀

A

∈

M

n

⁢

(

F

)

:

φ

⁢

(

A

)

=

P

⁢

A

⊤

⁢

Q

.

The original proof

[

D

]

of the nontrivial

implication

(i)

⇒

(ii) is based on the

fundamental theorem of projective geometry

.

References

D

J. Dieudonné, Sur une généralisation du groupe

orthogonal

à quatre

variables

,

Arch. Math.

1

: 282–287 (1949).

Title

Dieudonné

theorem

on linear preservers of the

singular matrices

Canonical name

DieudonneTheoremOnLinearPreserversOfTheSingularMatrices

Date of creation

2013-03-22 19:19:49

Last modified on

2013-03-22 19:19:49

Owner

kammerer (26336)

Last modified by

kammerer (26336)

Numerical id

8

Author

kammerer (26336)

Entry type

Theorem

Classification

msc 15A15

Classification

msc 15A04

Related topic

FundamentalTheoremOfProjectiveGeometry

Related topic

FrobeniusTheoremOnLinearDeterminantPreservers

Generated on Fri Feb 9 18:24:04 2018 by

LaTeXML
example of Cramer’s rule

example of Cramer’s rule

Say we want to solve the system

3

⁢

x

+

2

⁢

y

+

z

-

2

⁢

w

=

4

2

⁢

x

-

y

+

2

⁢

z

-

5

⁢

w

=

15

4

⁢

x

+

2

⁢

y

-

5

⁢

w

=

1

3

⁢

x

-

2

⁢

z

-

4

⁢

w

=

1

.

The associated matrix is

(

3

2

1

-

2

2

-

1

2

-

5

4

2

0

-

1

3

0

-

2

-

4

)

whose

determinant

is

Δ

=

-

65

. Since the determinant is non-zero, we can use Cramer’s rule. To obtain the value of the

k

-th variable, we replace the

k

-th column of the matrix above by the

column vector

(

4

15

1

1

)

,

the determinant of the obtained matrix is divided by

Δ

and the resulting value is the wanted solution.

So

x

=

Δ

1

Δ

=

|

4

2

1

-

2

15

-

1

2

-

5

1

2

0

-

1

1

0

-

2

-

4

|

-

65

=

-

65

-

65

=

1

y

=

Δ

2

Δ

=

|

3

4

1

-

2

2

15

2

-

5

4

1

0

-

1

3

1

-

2

-

4

|

-

65

=

130

-

65

=

2

z

=

Δ

3

Δ

=

|

3

2

4

-

2

2

-

1

15

-

5

4

2

1

1

3

0

1

-

4

|

-

65

=

-

195

-

65

=

3

w

=

Δ

4

Δ

=

|

3

2

1

4

2

-

1

2

15

4

2

0

1

3

0

-

2

1

|

-

65

=

65

-

65

=

-

1

Title

example of Cramer’s rule

Canonical name

ExampleOfCramersRule

Date of creation

2013-03-22 12:50:33

Last modified on

2013-03-22 12:50:33

Owner

drini (3)

Last modified by

drini (3)

Numerical id

7

Author

drini (3)

Entry type

Example

Classification

msc 15A15

Generated on Fri Feb 9 18:24:06 2018 by

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Hadamard product

Hadamard product

Definition

Suppose

A

=

(

a

i

⁢

j

)

and

B

=

(

b

i

⁢

j

)

are two

n

×

m

-matrices
with entries in some field. Then their

Hadamard product

is
the entry-wise

product

of

A

and

B

, that is,
the

n

×

m

-matrix

A

∘

B

whose

(

i

,

j

)

th entry is

a

i

⁢

j

⁢

b

i

⁢

j

.

Properties

Suppose

A

,

B

,

C

are matrices of the same size and

λ

is a scalar. Then

A

∘

B

=

B

∘

A

,

A

∘

(

B

+

C

)

=

A

∘

B

+

A

∘

C

,

A

∘

(

λ

⁢

B

)

=

λ

⁢

(

A

∘

B

)

,

•

If

A

,

B

are

diagonal matrices

, then

A

∘

B

=

A

⁢

B

.

•

(

Oppenheim inequality

)

[

2

]

: If

A

,

B

are

positive definite matrices

, and

(

a

i

⁢

i

)

are the

diagonal

entries of

A

, then

det

⁡

A

∘

B

≥

det

⁡

B

⁢

∏

a

i

⁢

i

with equality if and only if

A

is a diagonal matrix.

Remark

There is also a Hadamard product for two power series: Then the
Hadamard product of

∑

i

=

1

∞

a

i

and

∑

i

=

1

∞

b

i

is

∑

i

=

1

∞

a

i

⁢

b

i

.

References

1

R. A. Horn, C. R. Johnson,

Topics in Matrix Analysis

,
Cambridge University Press, 1994.

2

V.V. Prasolov,

Problems and Theorems in

Linear Algebra

,

American Mathematical Society

, 1994.

3

B. Mond, J. E. Pecaric,

Inequalities for the Hadamard product of matrices

,
SIAM Journal on Matrix Analysis and Applications,
Vol. 19, Nr. 1, pp. 66-70.

http://epubs.siam.org/sam-bin/dbq/article/30295

(link)

Title

Hadamard product

Canonical name

HadamardProduct

Date of creation

2013-03-22 14:15:28

Last modified on

2013-03-22 14:15:28

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

8

Author

bbukh (348)

Entry type

Definition

Classification

msc 15A15

Defines

Oppenheim inequality

Generated on Fri Feb 9 18:24:09 2018 by

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matrix resolvent properties

matrix resolvent properties

The matrix

resolvent

norm for a complex-valued

s

is related to the proximity of such value to the

spectrum

of

A

; more precisely, the following simple yet meaningful

property

holds:

∥

R

A

⁢

(

s

)

∥

≥

1

dist

⁢

(

s

,

σ

A

)

,

where

∥

.

∥

is any self

consistent

matrix norm

,

σ

A

is the spectrum of

A

and the

distance

between a complex point and the discrete set of the

eigenvalues

λ

i

is defined as

dist

⁢

(

s

,

σ

A

)

=

min

1

≤

i

≤

n

⁡

|

s

-

λ

i

|

.

From this fact it comes immediately, for any

1

≤

i

≤

n

,

lim

s

→

λ

i

⁡

∥

R

A

⁢

(

s

)

∥

=

+

∞

.

Proof.

Let (

λ

i

,

𝐯

) be an eigenvalue-eigenvector pair of

A

; then

(

s

⁢

I

-

A

)

⁢

v

=

s

⁢

v

-

A

⁢

v

=

(

s

-

λ

i

)

⁢

v

which shows

(

s

-

λ

i

)

to be an eigenvalue of

(

s

⁢

I

-

A

)

;

(

s

-

λ

i

)

-

1

is then an eigenvalue of

(

s

⁢

I

-

A

)

-

1

and , since for any self consistent norm

|

λ

|

≤

∥

A

∥

, we have:

max

1

≤

i

≤

n

⁡

1

|

s

-

λ

i

|

≤

∥

(

s

⁢

I

-

A

)

-

1

∥

whence

∥

(

s

⁢

I

-

A

)

-

1

∥

≥

1

min

1

≤

i

≤

n

⁡

|

s

-

λ

i

|

=

1

dist

⁢

(

s

,

σ

A

)

.

∎

Title

matrix resolvent properties

Canonical

name

MatrixResolventProperties

Date of creation

2013-03-22 15:33:52

Last modified on

2013-03-22 15:33:52

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

15

Author

Andrea Ambrosio (7332)

Entry type

Result

Classification

msc 15A15

Generated on Fri Feb 9 18:24:38 2018 by

LaTeXML
minor (of a matrix)

minor (of a matrix)

Given an

n

×

m

matrix

A

with entries

a

i

⁢

j

, a

minor

of

A

is
the

determinant

of a smaller matrix formed from its entries by selecting
only some of the rows and columns. Let

K

=

{

k

1

,

k

2

,

…

,

k

p

}

and

L

=

{

l

1

,

l

2

,

…

,

l

p

}

be subsets of

{

1

,

2

,

…

,

n

}

and

{

1

,

2

,

…

,

m

}

, respectively. The indices are chosen such that

k

1

<

k

2

<

⋯

<

k

p

and

l

1

<

l

2

<

⋯

<

l

p

. The

p

-th
order minor defined by

K

and

L

is the following determinant

A

⁢

(

k

1

k

2

⋯

k

p

l

1

l

2

⋯

l

p

)

=

|

a

k

1

⁢

l

1

a

k

1

⁢

l

2

⋯

a

k

1

⁢

l

p

a

k

2

⁢

l

1

a

k

2

⁢

l

2

⋯

a

k

2

⁢

l

p

⋮

⋮

⋱

⋮

a

k

p

⁢

l

1

a

k

p

⁢

l

2

⋯

a

k

p

⁢

k

p

|

.

If

p

exceeds either

m

or

n

, then the minor is
automatically zero. When

p

=

m

=

n

, the minor is simply the determinant
of the matrix. If

K

=

L

, then the minor is called

principal

.
The word

minor

may also refer to just the matrix formed from
the selected rows and columns, not necessarily its determinant. The precise
meaning is usually clear from

context

.

There does not seem to be a standard notation for matrix minors.
Another possible notation is

[

A

]

K

,

L

.

Some authors reserve the term

minor

for the case when only one
row and one column are removed. This use is in

conjunction

with the
concept of a

cofactor

(

http://planetmath.org/LaplaceExpansion

)

.

Title

minor (of a matrix)

Canonical name

MinorofAMatrix

Date of creation

2013-03-22 14:07:01

Last modified on

2013-03-22 14:07:01

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 15A15

Related topic

LaplaceExpansion

Related topic

CauchyBinetFormula

Defines

principal minor

Defines

cofactor

Generated on Fri Feb 9 18:24:41 2018 by

LaTeXML
n’th derivative of a determinant

n’th derivative of a determinant

Let

A

=

(

a

i

,

j

)

be a

d

×

d

matrix whose entries are

real functions

of

t

. Then,

d

n

d

⁢

t

n

⁢

det

⁡

(

A

)

=

∑

n

1

+

⋯

+

n

d

=

n

(

n

n

1

,

n

2

,

…

,

n

d

)

⁢

∑

π

∈

S

d

sgn

⁡

(

π

)

⁢

∏

i

=

1

d

d

n

i

d

⁢

t

n

i

⁢

a

i

,

π

⁢

(

i

)

=

∑

n

1

+

⋯

+

n

d

=

n

(

n

n

1

,

n

2

,

…

,

n

d

)

⁢

det

⁡

(

d

n

1

d

⁢

t

n

1

⁢

a

1

,

1

d

n

1

d

⁢

t

n

1

⁢

a

1

,

2

⋯

d

n

1

d

⁢

t

n

1

⁢

a

1

,

d

⋮

⋮

⋮

d

n

d

d

⁢

t

n

d

⁢

a

d

,

1

d

n

d

d

⁢

t

n

d

⁢

a

d

,

2

⋯

d

n

d

d

⁢

t

n

d

⁢

a

d

,

d

)

where

(

n

n

1

,

n

2

,

…

,

n

r

)

is the

multinomial coefficient

,

S

d

is the

symmetric group

of

permutations

and

sgn

⁡

(

π

)

is the sign of a
permutation

π

.

Title

n’th derivative of a

determinant

Canonical name

NthDerivativeOfADeterminant

Date of creation

2013-03-22 14:30:25

Last modified on

2013-03-22 14:30:25

Owner

GeraW (6138)

Last modified by

GeraW (6138)

Numerical id

5

Author

GeraW (6138)

Entry type

Result

Classification

msc 15A15

Related topic

GeneralizedLeibnizRule

Related topic

MultinomialTheorem

Related topic

DerivativeOfMatrix

Generated on Fri Feb 9 18:24:43 2018 by

LaTeXML
Pfaffian

Pfaffian

The

Pfaffian

is an analog of the

determinant

that is defined only for a

2

⁢

n

×

2

⁢

n

antisymmetric matrix

. It is a

polynomial

of the polynomial ring in elements of the matrix, such that its square is equal to the determinant of the matrix.

The Pfaffian is applied in the generalized Gauss-Bonnet theorem.

Examples

P

⁢

f

⁢

[

0

a

-

a

0

]

=

a

,

P

⁢

f

⁢

[

0

a

b

c

-

a

0

d

e

-

b

-

d

0

f

-

c

-

e

-

f

0

]

=

a

⁢

f

-

b

⁢

e

+

d

⁢

c

.

Standard definition

Let

A

=

[

0

a

1

,

2

…

a

1

,

2

⁢

n

-

a

1

,

2

0

…

a

2

,

2

⁢

n

⋮

⋮

⋮

⋮

-

a

2

⁢

n

,

1

-

a

2

⁢

n

,

2

…

0

]

.

Let

Π

be the set of all

partition

of

{

1

,

2

,

…

,

2

⁢

n

}

into pairs of elements

α

∈

Π

, can be represented as

α

=

{

(

i

1

,

j

1

)

,

(

i

2

,

j

2

)

,

…

,

(

i

n

,

j

n

)

}

with

i

k

<

j

k

and

i

1

<

i

2

<

⋯

<

i

n

, let

π

=

[

1

2

3

4

…

2

⁢

n

i

1

j

1

i

2

j

2

…

j

n

]

be a corresponding

permutation

and let us define

s

⁢

g

⁢

n

⁢

(

α

)

to be the

signature

of a permutation

π

; clearly it depends only on the partition

α

and not on the particular choice of

π

.
Given a partition

α

as above let us set

a

α

=

a

i

1

,

j

1

⁢

a

i

2

,

j

2

⁢

…

⁢

a

i

n

,

j

n

,

then we can define the

Pfaffian

of

A

as

P

⁢

f

⁢

(

A

)

=

∑

α

∈

Π

s

⁢

g

⁢

n

⁢

(

α

)

⁢

a

α

.

Alternative definition

One can

associate

to any

antisymmetric

2

⁢

n

×

2

⁢

n

matrix

A

=

{

a

i

⁢

j

}

a bivector
:

ω

=

∑

i

<

j

a

i

⁢

j

⁢

e

i

∧

e

j

in a basis

{

e

1

,

e

2

,

…

,

e

2

⁢

n

}

of

ℝ

2

⁢

n

, then

ω

n

=

n

!

⁢

P

⁢

f

⁢

(

A

)

⁢

e

1

∧

e

2

∧

⋯

∧

e

2

⁢

n

,

where

ω

n

denotes

exterior product

of

n

copies of

ω

.

Identities

For any antisymmetric

2

⁢

n

×

2

⁢

n

matrix

A

’ and any

2

⁢

n

×

2

⁢

n

matrix

B

P

⁢

f

⁢

(

A

)

2

=

det

⁡

(

A

)

P

⁢

f

⁢

(

B

⁢

A

⁢

B

T

)

=

det

⁡

(

B

)

⁢

P

⁢

f

⁢

(

A

)

Title

Pfaffian

Canonical name

Pfaffian

Date of creation

2013-03-22 14:22:13

Last modified on

2013-03-22 14:22:13

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

26

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 15A15

Generated on Fri Feb 9 18:24:48 2018 by

LaTeXML
proof of block determinants

proof of block determinants

If

A

-

1

exists, then

(

A

B

C

D

)

=

(

I

O

C

⁢

A

-

1

I

)

⁢

(

A

B

O

D

-

C

⁢

A

-

1

⁢

B

)

.

So

det

⁡

(

A

B

C

D

)

=

det

⁡

(

I

O

C

⁢

A

-

1

I

)

⁢

det

⁡

(

A

B

O

D

-

C

⁢

A

-

1

⁢

B

)

.

Each of the first matrices in the

decompositions

are triangular. Hence their

determinants

equal

1

. This means that the determinant of the original matrix equals the determinant of either of the second matrices in the decomposition. Therefore

det

⁡

(

A

B

C

D

)

=

det

⁡

(

A

)

⁢

det

⁡

(

D

-

C

⁢

A

-

1

⁢

B

)

.

The second formula follows by using a

similar

trick.

Title

proof of block determinants

Canonical name

ProofOfBlockDeterminants

Date of creation

2013-03-22 15:27:49

Last modified on

2013-03-22 15:27:49

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

5

Author

georgiosl (7242)

Entry type

Proof

Classification

msc 15A15

Generated on Fri Feb 9 18:24:50 2018 by

LaTeXML
proof of cofactor expansion

proof of cofactor expansion

Let

M

∈

m

⁢

a

⁢

t

N

⁢

(

K

)

be a

n

×

n

-matrix with entries from a

commutative

field

K

. Let

e

1

,

…

,

e

n

denote the vectors of the canonical basis of

K

n

. For the proof we need the following

Lemma:

Let

M

i

⁢

j

*

be the matrix generated by replacing the

i

-th
row of

M

by

e

j

. Then

det

⁡

M

i

⁢

j

*

=

(

-

1

)

i

+

j

⁢

det

⁡

M

i

⁢

j

where

M

i

⁢

j

is the

(

n

-

1

)

×

(

n

-

1

)

-matrix obtained from

M

by removing
its

i

-th row and

j

-th column.

Proof.

By adding appropriate of the

i

-th row of

M

i

⁢

j

*

to its remaining rows we obtain a matrix with 1 at position

(

i

,

j

)

and 0 at
positions

(

k

,

j

)

(

k

≠

i

). Now we apply the

permutation

(

12

)

∘

(

23

)

∘

…

∘

(

(

i

-

1

)

⁢

i

)

to rows and

(

12

)

∘

(

23

)

∘

…

∘

(

(

j

-

1

)

⁢

j

)

to columns of the matrix. The matrix now looks like this:

•

Row/column 1 is the vector

e

1

;

•

under row 1 and right of column 1 is the matrix

M

i

⁢

j

.

Since the

determinant

has changed its sign

i

+

j

-

2

times, we have

det

⁡

M

i

⁢

j

*

=

(

-

1

)

i

+

j

⁢

det

⁡

M

i

⁢

j

.

Note also that only those permutations

π

∈

S

n

are for the

computation

of the determinant of

M

i

⁢

j

*

where

π

⁢

(

i

)

=

j

.
∎

Now we start out with

det

⁡

M

=

∑

π

∈

S

n

sgn

⁢

π

⁢

(

∏

j

=

1

n

m

j

⁢

π

⁢

(

j

)

)

=

∑

k

=

1

n

m

i

⁢

k

⁢

(

∑

π

∈

S

n

∣

π

(

i

)

=

k

sgn

⁢

π

⁢

(

∏

1

≤

j

≤

i

m

j

⁢

π

⁢

(

j

)

)

⋅

1

⋅

(

∏

i

≤

j

≤

n

m

j

-

π

⁢

(

j

)

)

)

.

From the previous lemma, it follows that the associated with

M

i

⁢

k

is the determinant of

M

i

⁢

j

*

. So we have

det

⁡

M

=

∑

k

=

1

n

M

i

⁢

k

⁢

(

(

-

1

)

i

+

k

⁢

det

⁡

M

i

⁢

k

)

.

Title

proof of cofactor expansion

Canonical name

ProofOfCofactorExpansion

Date of creation

2013-03-22 13:22:08

Last modified on

2013-03-22 13:22:08

Owner

Thomas Heye (1234)

Last modified by

Thomas Heye (1234)

Numerical id

13

Author

Thomas Heye (1234)

Entry type

Proof

Classification

msc 15A15

Synonym

Laplace expansion

Generated on Fri Feb 9 18:24:53 2018 by

LaTeXML
proof of Cramer’s rule

proof of Cramer’s rule

Since

det

⁡

(

A

)

≠

0

, by

properties

of the

determinant

we know that

A

is

invertible

.

We claim that this implies that the equation

A

⁢

x

=

b

has a unique solution.
Note that

A

-

1

⁢

b

is a solution since

A

⁢

(

A

-

1

⁢

b

)

=

(

A

⁢

A

-

1

)

⁢

b

=

b

, so we know that a solution exists.

Let

s

be an arbitrary solution to the equation, so

A

⁢

s

=

b

. But then

s

=

(

A

-

1

⁢

A

)

⁢

s

=

A

-

1

⁢

(

A

⁢

s

)

=

A

-

1

⁢

b

, so we see that

A

-

1

⁢

b

is the only solution.

For each

integer

i

,

1

≤

i

≤

n

, let

a

i

denote the

i

th column of

A

, let

e

i

denote the

i

th column of the

identity matrix

I

n

, and let

X

i

denote the matrix obtained from

I

n

by replacing column

i

with the

column vector

x

.

We know that for any matrices

A

,

B

that the

k

th column of the product

A

⁢

B

is simply the product of

A

and the

k

th column of

B

. Also observe that

A

⁢

e

k

=

a

k

for

k

=

1

,

…

,

n

. Thus, by

multiplication

, we have:

A

⁢

X

i

=

A

⁢

(

e

1

,

…

,

e

i

-

1

,

x

,

e

i

+

1

,

…

,

e

n

)

=

(

A

⁢

e

1

,

…

,

A

⁢

e

i

-

1

,

A

⁢

x

,

A

⁢

e

i

+

1

,

…

,

A

⁢

e

n

)

=

(

a

1

,

…

,

a

i

-

1

,

b

,

a

i

+

1

,

…

,

a

n

)

=

M

i

Since

X

i

is

I

n

with column

i

replaced with

x

, computing the determinant of

X

i

with

cofactor expansion

gives:

det

⁡

(

X

i

)

=

(

-

1

)

(

i

+

i

)

⁢

x

i

⁢

det

⁡

(

I

n

-

1

)

=

1

⋅

x

i

⋅

1

=

x

i

Thus by the

multiplicative

property of the determinant,

det

⁡

(

M

i

)

=

det

⁡

(

A

⁢

X

i

)

=

det

⁡

(

A

)

⁢

det

⁡

(

X

i

)

=

det

⁡

(

A

)

⁢

x

i

and so

x

i

=

det

⁡

(

M

i

)

det

⁡

(

A

)

as required.

Title

proof of Cramer’s rule

Canonical

name

ProofOfCramersRule

Date of creation

2013-03-22 13:03:24

Last modified on

2013-03-22 13:03:24

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

11

Author

rmilson (146)

Entry type

Proof

Classification

msc 15A15

Generated on Fri Feb 9 18:24:55 2018 by

LaTeXML