Matrix theory basics 1.1 second-and third-order Determinant

Section 1 second-and third-order Determinant

Before introducing the concept of determining factors, we first construct a math: place the four numbers on the four corners of a square, and add two vertical bars. That is to say, this toy corresponds to a result: the difference between the product of numbers on two diagonal lines. That is

For example

The diagonal line in the direction is called the main diagonal line, and the diagonal line in the direction is called the secondary diagonal line.

Definition 1 4 number is called a second-order Determinant; the row is called the first row, which is recorded as (r comes from the English row), and the column is called the second column, which is recorded as (c comes from the English column ), because it has two rows and two columns, it is called the Second-Order Determinant and is the element of the first column of the Second row.

It is generally used to represent the elements in column j of row I. column I is the row mark and column j is the column mark.

It can be described that the corresponding value of the Second-Order Determinant is equal to the difference obtained from the product of the two elements on the primary diagonal minus the product of the two elements on the secondary diagonal. This calculation rule is called the Diagonal rule.

The purpose of this toy is:

Solving Equations

Using the elimination method, first remove the item, equation (2 )'A11, equation (1 )'A21de

(3)-(4), get

Then remove the item, equation (2 )'A12, equation (1 )'A22nd

(5)-(6), get

We found that the law is: if we remember the coefficient determinant of the equations,

It is to replace the first column of D with the constant term;

Replace the second column of D with a constant.

If D is less than 0, the equations are exactly :,

This law is called the Cramer theorem.

Example 1 solving Binary Linear Equations

Solution :,

,

,

Therefore ,.

Likewise, the diagonal line rule can be used to define the level 3 determinant as follows:

The product of the three elements on the three main diagonal lines is preceded by a positive number;

The product of three elements on the diagonal lines of three pairs. A negative sign is added to the front.

Example 2 Calculate the Level 3 Determinant

Solution:D= 1 × 2 × 2 + 3 × 1 × 1 + 3 × 1 × (-1)-1 × 2 × 3-(-1) ×1 × 1-2 × 1 × 3 =-7

D1 = 6 × 2 × 2 + 4 × 1 × 1 + 11 × 1 × (-1)-1 × 2 × 11-(-1) ×1 × 6-2 × 1 × 4 =-7

D2 = 1 × 4 × 2 + 3 × 11 × 1 + 3 × 6 × (-1)-1 × 4 × 3-(-1) × 11 × 1-2 × 6 × 3 =-14

D3 = 1 × 2 × 11 + 3 × 1 × 6 + 3 × 1 × 4-6 × 2 × 3-4 × 1 × 1-11 × 1 × 3 = -- 21

Actually,D,D1,D2,D3 from linear equations