cofactor expansion

cofactor expansion

Let

M

be an

n

×

n

matrix with entries

M

i

⁢

j

that are

elements of

a

commutative ring

.
Let

m

i

⁢

j

denote the

determinant

of the

(

n

-

1

)

×

(

n

-

1

)

submatrix

obtained by deleting
row

i

and column

j

of

M

, and let

C

i

⁢

j

=

(

-

1

)

i

+

j

⁢

m

i

⁢

j

.

The

subdeterminants

m

i

⁢

j

are called the

minors

of

M

, and the

C

i

⁢

j

are called the

cofactors

.

We have the following useful formulas for the cofactors of a matrix.
First, if we regard

det

⁡

M

as a

polynomial

in the entries

M

i

⁢

j

, then we may write

C

i

⁢

j

=

∂

⁡

M

∂

⁡

M

i

⁢

j

(1)

Second, we may regard the determinant of

M

=

(

M

1

,

…

,

M

n

)

as a

multi-linear

,

skew-symmetric

function

of its columns:

det

⁡

M

=

det

⁡

(

M

1

,

…

,

M

n

)

.

This point of view leads to the following formula:

C

i

⁢

j

=

det

⁡

(

M

1

,

…

,

M

j

^

,

𝐞

i

,

…

,

M

n

)

,

(2)

where the notation indicates that column

j

has been replaced by the

i

th standard vector.

As a consequence, we obtain the following

representation

of the determinant in terms of
cofactors:

det

⁡

(

M

)

=

det

⁡

(

M

1

,

…

,

M

1

⁢

j

⁢

𝐞

1

+

⋯

+

M

n

⁢

j

⁢

𝐞

n

,

…

,

M

n

)

=

∑

i

=

1

n

M

i

⁢

j

⁢

C

i

⁢

j

,

j

=

1

,

…

,

n

.

The above

identity

is often called the cofactor expansion of the determinant along column

j

.
If we regard the determinant as a multi-linear, skew-symmetric function of

n

row-vectors, then we obtain the
analogous cofactor expansion along a row:

det

⁡

(

M

)

=

∑

i

=

1

n

M

j

⁢

i

⁢

C

j

⁢

i

.

Example.

Consider a general

3

×

3

determinant

|

a

1

a

2

a

3

b

1

b

2

b

3

c

1

c

2

c

3

|

=

a

1

⁢

b

2

⁢

c

3

+

a

2

⁢

b

3

⁢

c

1

+

a

3

⁢

b

1

⁢

c

2

-

a

1

⁢

b

3

⁢

c

2

-

a

3

⁢

b

2

⁢

c

1

-

a

2

⁢

b

1

⁢

c

3

.

The above can equally well be expressed as a cofactor expansion along
the first row:

|

a

1

a

2

a

3

b

1

b

2

b

3

c

1

c

2

c

3

|

=

a

1

⁢

|

b

2

b

3

c

2

c

3

|

-

a

2

⁢

|

b

1

b

3

c

1

c

3

|

+

a

3

⁢

|

b

1

b

2

c

1

c

2

|

=

a

1

⁢

(

b

2

⁢

c

3

-

b

3

⁢

c

2

)

-

a

2

⁢

(

b

1

⁢

c

3

-

b

3

⁢

c

1

)

+

a

3

⁢

(

b

1

⁢

c

2

-

b

2

⁢

c

1

)

;

or along the second column:

|

a

1

a

2

a

3

b

1

b

2

b

3

c

1

c

2

c

3

|

=

-

a

2

⁢

|

b

1

b

3

c

1

c

3

|

+

b

2

⁢

|

a

1

a

3

c

1

c

3

|

-

c

2

⁢

|

a

1

a

3

b

1

b

3

|

=

-

a

2

⁢

(

b

1

⁢

c

3

-

b

3

⁢

c

1

)

+

b

2

⁢

(

a

1

⁢

c

3

-

a

3

⁢

c

1

)

-

c

2

⁢

(

a

1

⁢

b

3

-

a

3

⁢

b

1

)

;

or indeed as four other such expansion corresponding to rows 2 and 3,
and columns 1 and 3.

Title

cofactor expansion

Canonical name

CofactorExpansion

Date of creation

2013-03-22 12:01:07

Last modified on

2013-03-22 12:01:07

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

17

Author

rmilson (146)

Entry type

Theorem

Classification

msc 15A15

Synonym

Laplace expansion

Synonym

cofactor

Synonym

minor

Synonym

subdeterminant

Related topic

SarrusRule

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