Matrix’s determinant

Signing:

singular

Theorem: ,

Proof: without proof.

Definition:

A and B matrix’s are similar if T regular :

Theorem: the similarity is an equivalent relation.

Proof: it comes from the theorem.

Theorem: every square matrix is similar with a triangle one.

Proof: without proof.

Determinant determination:

n=2 :

=

n=3:

=

We can count it like it because of the permutation.

Theorem:

1.

2.

3.The determinants of similar matrix’s are the same.

Proof:

1.

2. ,

3.

Q.E.D.

Determinant of a triangle matrix: the product of the elements in the main counter-diagonal.

La Place endorsement

Definition:

A matrix, then

matrix is created from ‘A’ by leaving its “i” row and “j” column.

Theorem:

Proof: without proof.

column

endorsement after the j-th column.

For example:

j=2

For example:

j=1

Definition:

Definition:

For example:

Theorem: A is a regular one (quadratic matrix), then

Proof:

(theorem)

a, La Place endorsement after the column.

b,

is the same as the determinant of the matrix, created by the changing of the row to the row.

and row are the same

the matrix is a singular one

,

3. /multiplication from left

/

Q.E.D

For example: