determinant

determinant

Overview

The

determinant

is an algebraic operation that transforms a

square matrix

M

into a scalar. This

operation

has many useful and
important properties. For example, the determinant is zero if and
only the matrix

M

is

singular

(no

inverse

exists). The determinant
also has an important geometric

interpretation

as the area of a

parallelogram

, and more generally as the volume of a
higher-dimensional

parallelepiped

.

The notion of determinant predates matrices and

linear
transformations

. Originally, the determinant was a number associated
to a system of

n

linear equations

in

n

variables. This number
“determined” whether the system possessed a unique

solution

. In
this sense, two-by-two determinants were considered by Cardano at the
end of the 16th century and ones of arbitrary size (see the definition
below) by Leibniz about 100 years later.

Definition

Let

M

be an

n

×

n

matrix with entries

M

i

⁢

j

that are
elements of a given field

1

1

Most scientific and geometric
applications deal with matrices made up of real or

complex numbers

.
However, the determinant of a matrix over any field is well defined
sense and has all the properties of the more conventional
determinant. Indeed, many properties of the determinant remain
valid for matrices with entries in a

commutative ring

.

. The
determinant of

M

, or

det

⁡

M

for short, is the scalar quantity

det

⁡

M

=

|

M

11

M

12

…

M

1

⁢

n

M

21

M

22

…

M

2

⁢

n

⋮

⋮

⋱

⋮

M

n

⁢

1

M

n

⁢

2

…

M

n

⁢

n

|

=

∑

π

∈

S

n

sgn

⁢

(

π

)

⁢

M

1

⁢

π

1

⁢

M

2

⁢

π

2

⁢

⋯

⁢

M

n

⁢

π

n

.

(1)

The index

π

in the above sum varies over all the

permutations

of

{

1

,

…

,

n

}

(i.e., the elements of the

symmetric group

S

n

.)
Hence, there are

n

!

terms in the defining sum of the determinant.
The symbol

sgn

⁡

(

π

)

denotes the parity of the
permutation; it is

±

1

according to whether

π

is an even or

odd permutation

. Using the

Einstein summation convention

one can also
express the above definition as

det

M

=

ϵ

π

1

⁢

π

2

⁢

…

⁢

π

n

M

π

1

M

π

2

1

⋯

2

M

π

n

,

n

(2)

where we’ve raised the first index so that

M

i

=

j

M

i

⁢

j

, and
where

ϵ

π

1

⁢

…

⁢

π

n

=

sgn

⁡

(

π

)

is known as the

Levi-Civita permutation symbol

.

By way of example, the determinant of a

2

×

2

matrix is given by

|

M

11

M

12

M

21

M

22

|

=

M

11

⁢

M

22

-

M

12

⁢

M

21

,

There are six permutations of the numbers

1

,

2

,

3

, namely

1

⁢

2

+

⁢

3

,

2

⁢

3

+

⁢

1

,

3

⁢

1

+

⁢

2

,

1

⁢

3

-

⁢

2

,

3

⁢

2

-

⁢

1

,

2

⁢

1

-

⁢

3

;

the overset sign indicates the permutation’s

signature

. Accordingly,
the

3

×

3

deterimant is a sum of the following

6

terms:

|

M

11

M

12

M

13

M

21

M

22

M

23

M

31

M

32

M

33

|

=

M

11

⁢

M

22

⁢

M

33

+

M

12

⁢

M

23

⁢

M

31

+

M

13

⁢

M

21

⁢

M

32

-

M

11

⁢

M

23

⁢

M

32

-

M

13

⁢

M

22

⁢

M

31

-

M

12

⁢

M

21

⁢

M

33

Remarks and important properties

1.

The determinant operation converts

matrix multiplication

into

scalar multiplication

;

det

⁡

(

A

⁢

B

)

=

det

⁡

(

A

)

⁢

det

⁡

(

B

)

,

where

A

,

B

are square matrices of the same size.

2.

The determinant operation is

multi-linear

, and

anti-symmetric

with respect to the matrix’s rows and columns. See the
multi-linearity attachment for more details.

3.

The determinant of a

lower triangular

, or an

upper triangular
matrix

is the

product

of the

diagonal

entries, since all the other summands
in (1) are zero.

4.

Similar matrices

(

http://planetmath.org/SimilarMatrix

)
have the same determinant. To be more precise,
let

A

and

X

be square matrices with

X

invertible

. Then,

det

⁡

(

X

⁢

A

⁢

X

-

1

)

=

det

⁡

(

A

)

.

In particular, if we let

X

be the matrix representing a

change of
basis

, this shows that the determinant is

independent

of the basis.
The same is true of the trace of a matrix. In fact, the whole

characteristic polynomial

of an

endomorphism

is

definable

without
using a basis or a matrix, and it turns out that the determinant and
trace are two of its

coefficients

.

5.

The determinant of a matrix

A

is zero if and only if

A

is
singular; that is, if there exists a non-trivial solution to the
homogeneous equation

A

⁢

𝐱

=

𝟎

.

6.

The

transpose

operation does not change the determinant:

det

⁡

A

T

=

det

⁡

A

.

7.

The determinant of a

diagonalizable

transformation

is equal to the product
of its

eigenvalues

, counted with

multiplicities

.

8.

The determinant is

homogeneous of degree

n

. This means that

det

⁡

(

k

⁢

M

)

=

k

n

⁢

det

⁡

M

,

k

⁢

is a scalar.

Title

determinant

Canonical name

Determinant

Date of creation

2013-03-22 12:33:07

Last modified on

2013-03-22 12:33:07

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

24

Author

rmilson (146)

Entry type

Definition

Classification

msc 15A15

Related topic

LaplaceExpansion

Related topic

Permanent

Related topic

GeneralizedRuizsIdentity

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