Matrix Theory Basis 1.4 nature of the determinant

Section 4 nature of the determinant

There are seven attributes of the determinant:

Order n determinant:

If you change the row of D to A column, the new criterion is as follows:

, DeterminantDT(Or D ') is called the deciding factorDThe transpose deciding factor.

Note: The transpose deciding factor can also be seen as the result of turning 180 ° with the primary diagonal line as the axis.

Property 1 determine the D = DT

Proof :,

When n = 2, the conclusion is clearly true, that is

N-1, the conclusion is true, that is, the rank-1 determinant is equal to its transpose determinant, And the rank-n determinant D is expanded according to the first row.

Deploy the DT of order n according to the first column.

So the n-order Determinant D = DT

From the Nature 1 of the determinant, we can see that the row of the determinant is the same as the column, and the row has the same nature for the column, and vice versa.

Property 2 if a row (or column) in the determinant is zero, the value of this determinant is equal to zero.

Note: Expand the Determinant by this row (or column.

Property 3 any two rows (or two columns) in the determinant are interchange locations, and the value of the determinant is changed to numbers.

Proof: After the positions of the first and third rows are swapped, the determinant changes

Expand D according to the first line.

Expand D1 by the third line.

This property is also true for the rank-n determinant.

Inference: If two rows (columns) of the determinant are exactly the same, the determinant is equal to zero.

(Exchange the two rows (columns) determine the D to D1, known by the nature of 2,-D = D1, because the exchange of the two rows (columns) are the same, so

D = D1, So-D = D, D = 0

All elements in a row (column) of attribute 4 are multiplied by the same number.λ, Equal to the numberλTake this determinant. On the contrary, if all the elements in a row (column) of the determinant have a public factor, this public factor can be proposed from the determinant, that is

Note: If the two terminologies above are expanded by line I, the results are the same.

Inference: if there are two rows (columns) in the determinant that correspond proportionally, then this determinant is equal to zero.

Property 5: if each element in a row (column) of a determinant is the sum of two numbers, for exampleIThe elements of a row are the sum of two numbers:

,

ThenDEquals to the sum of the following two determining factors:

.

Note: Remember that the three determine factors are D, D1, D2, then

Property 6: multiply each element of a row (column) of the determinant by the same number and add it to the corresponding element of another row (Column). The determinant remains unchanged. That is

.

Note: Properties 5 and 4 are available. properties 6 are very important in the calculation of the determinant.

Property 7 determine the sum of the product of each element in each row (or column) and the algebraic Remainder of the element corresponding to the other row (or column) is equal to zero, that is

Note: The Order N determining factor is expanded by row J,

The following conclusion is obtained:

The above seven attributes are often used for processing and calculating the deciding factor. To express the conciseness, the following mark is introduced.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

For example,

Example 9 calculate the Determinant

Solution: using the nature of the determinant, convert d into an equal upper (lower) triangular determinant, and then write the result. This is a common method for calculating the determinant.

Note:

(1) Use Property 6 to convert all elements under a11 to zero first;

(2) convert all the elements under a22 to zero;

(3) Repeat the operation until it is converted into a triangular determinant;

(4) You can use the same processing method for a column to convert it into another type of triangular determinant and then evaluate it.

When determining the value of a determining factor, the common method is to expand by a row (column) to reduce the order, so as to simplify the determining factor until the result is obtained.

Example 10 calculate the Determinant

Solution:

We should be good at using two methods to find the value of the determinant:

1. Convert the result into a triangular determinant (four results)

2. Expand by a row or column (Select zero rows or columns )).

Example 11 calculate the Determinant

Solution: because the first column is proportional to the third column, D = 0.

Example 12 calculate the Determinant

Solution:

Example 13 calculate the Determinant

Solution: each row in D1 puts forward a public factor (-1 ).

So D1 = 0

D2 is expanded by the first line.

Example 14 calculate the Determinant

Solution:

Likewise

Example 15 calculate the third-level vandmonde Determinant

Solution:

Similarly, we can obtain the rank-n vandmonde.

Example 16 calculate the zero-block determinant of (m + n) Order

Solution: note,

| A | perform several ri + λ rj operations to convert them into the lower triangle determinant and set it

| B | perform several ci + λ cj operations, convert them to the lower triangle determinant, and set it

Apply all the operations on | A | to the first n rows of D, and then apply all the operations on | B | to the rear m column of D.

Likewise, we can see the values of the following three zero-block factors.

(1)

(2)

Note:

(3)

English version (1. (2) In the deciding factor D can be converted into the lower triangle deciding factor, using the previous conclusion, can be pushed)

2. Remember the four results.