How to calculate the algebraic remainder, how to find the algebraic coundrel

The specific solution steps of the algebraic remainder formula:First of all, the sum of the algebraic remainder of the first row is equal to a determinant obtained by replacing the elements of the first row of the original determinant with the number "1", and the sum of the algebraic remainder of the second row is equal to the determinant obtained by replacing the elements of the second row of the atomic determinant with the number "1", so through this law, we can see that the sum of the algebraic remainder of the nth row of the original determinant is also equal to the determinant obtained by converting all the elements in the nth row of the original determinant into the number "1". And the sum of all algebraic remainders is the sum of the n new determinants above.

When we encounter problems in daily calculations, we can directly multiply the focus array formed by multiple exchanges for each exchange by -1, or according to the sum of the first columns, the coefficient of the algebraic remainder formula is (-1) (5+1), and in the same case, the final result can be obtained when the remainder formula is carried out according to a certain row and a certain column.

What are the properties of algebraic remainders? According to the determinant a row (column) multiplied by the same number k, the result is ka, and the determinant a is equal to the other transposed determinants at (at is the nth column of the nth row of the determinant a), if the nth order determinant |αij|, you can get the determinant |αij|is the sum of two determinants. then the meta values on the remaining rows (columns) are sum |αij|It's exactly the same.

Algebraic codon formula is related to the concept of cocontinuation: algebraic coundit = corresponding coexistent formula [position coefficient] (either +1, or -1).

The sum of the algebraic remainders of the first line is equal to the determinant obtained by replacing all the elements of the first row of the original determinant with 1, the sum of the algebraic remainders of the second line of the original determinant is equal to the determinant obtained by replacing all the elements of the second line of the original determinant with 1, the sum of the algebraic remainders of the nth row of the original determinant is equal to the determinant obtained by replacing all the elements of the nth row of the original determinant with 1, and the sum of all algebraic remainders is the sum of the n new determinants above.

It is possible to form a diagonal array directly through several exchange rows, multiplying each exchange by a -1. Or according to the first column, the coefficient of the algebraic remainder is (-1) (5+1), because the subscript of 6 is 51, and the remainder is followed by a row or column.

Nature A row (or column) in determinant a is multiplied by the same number k, and the result is equal to ka.

The determinant a is equal to its transposed determinant at (the ith line of at is column ith of a).

If the nth order determinant |αij|a row (or column) in ; The determinant is |αij|is the sum of two determinants, the ith row (or column) of these two determinants, one is b1, b2 ,...，bn；The other is 1, 2,...，n；The meta on the remaining rows (or columns) is the same as |αij|of exactly the same.

Algebraic coundons are for an element of a determinant.

The solution method is to cross out the row and column where the element is located to form a lower-order determinant, and then find the value of this determinant; After solving, multiply by the sign of the position of this element, and the solution is (-1) (row of element + column of element).

Take a look at the image below:

In an nth-order determinant d, put the element aij (i,j=1,2,..After the row and column of n) are crossed out, the remaining (n-1) two elements form an n-1 determinant mij in the original order, which is called the coundant of the element aij, and mij with the symbol (-1) (i+j) is called the algebraic coundant of aij, which is denoted as aij=(-1) (i+j) mij.

Hello! In a determinant, the determinant formed by crossing out the j-column of the ith row and the remaining elements according to their original positions is called the courinator formula, which is denoted as mij, and the algebraic coundan aij is the coundant before multiplying 1 or minus 1, that is, aij=[(-1) (i+j)]mij. The Economic Mathematics team will help you solve the problem, please adopt it in time. Thank you!

The sum of all algebraic coundons is equal to the sum of all the elements of this adjoint matrix, just find its adjoint matrix, and then add the elements of the adjoint matrix to find it.

In the nth-order determinant, after the o-row and e-column where the element a i is located are crossed out, the remaining n-1 determinant is called the coremainant of element a i, which is denoted as m, and the coundit m is multiplied by the o+e power of -1 as a, and a is called the algebraic coremainant of element a. The algebraic remainder of an element a i has nothing to do with the element itself, only with the position of that element.

Algebraic remainder:

In the nth-order determinant, after the o-row and e-column where the element a i is located are crossed out, the remaining n-1 determinant is called the coremainant of element a i, which is denoted as m, and the coundit m is multiplied by the o+e power of -1 as a, and a is called the algebraic coremainant of element a.

The algebraic remainder of an element a i has nothing to do with the element itself, only with the position of that element.

Example: Example 1 in a fifth-order determinant.

, delineate the second row, the fourth row, and the second and third columns, and determine a second-order sub-determinant of d.

The corresponding coincidence m of a is: <>

The corresponding algebraic remainder of the sub-determinant a is:

In an nth-order determinant d, put the element aij (i,j=1,2,..After the row and column of n) are crossed out, the remaining (n-1) two elements form an n-1 determinant mij in the original order, which is called the coundant of the element aij, and mij with the symbol (-1) (i+j) is called the algebraic coundant of aij, which is denoted as aij=(-1) (i+j) mij.

a*t=at*

Yes, the adjoint matrix transposed by a is equal to the transpose of the adjoint matrix of a.

How to find the algebraic remainder.

Algebraic remainder:

It is extracted from the formula of the determinant, and its function is to reduce the n-order determinant to the n-order determinant. In the nth-order determinant, after the o-row and e-column where the element a i is located are crossed out, the remaining n-1 determinant is called the coremainant of element a i, which is denoted as m, and the coundit m is multiplied by the o+e power of -1 as a, and a is called the algebraic coremainant of element a.

In the n-order determinant, the n-1 order determinant formed by crossing out the elements in row i and column j where the yuan aij is located, and the remaining yuan does not change the original order is called the coexistant of the yuan aij.

Relationship: <>

The algebraic remainder itself is an n-1 order determinant, and it can continue to be an n-2 order determinant.

And so on, until the first-order determinant, the core idea is to convert a complex higher-order determinant into multiple simple lower-order determinants.

There are three differences between coexistent and algebraic couters: different references, different characteristics, and different uses.

First, the reference is different.

1. Coincidental formula: The lower the order of the determinant, the easier it is to calculate. Therefore, we naturally ask whether a higher-order determinant can be converted into a low-order determinant for ruler calculation.

2. Algebraic coundifier: In the nth-order determinant, after removing the other row of element A and the E column Trapping Feast I, the remaining n1-order determinant is called the coremainant of element A i.

Second, the characteristics are different.

1. Coincidental formula: The coremainant formula of a k-order sub-formula is the determinant of the (n k) (n k) matrix obtained by a after removing the rows and columns where the k-order sub-formula is located.

2. Algebraic cocontinuation: The algebraic coundron of element a i has nothing to do with the element itself, only with the position of the element.

Third, the use is different.

1. Coincidental formula: The transpose matrix is called the adjoint matrix of a. Adjoint matrices are similar to inverse matrices and can be used to compute the inverse matrix of a when a is reversible.

2. Algebraic coundifiers: When calculating the algebraic coundrons of an element, the first thing to do is not to ignore the algebraic notation of the cognostics. When calculating a linear combination of element cofactors for a row (or column), you can directly calculate each cofactor and then sum it.

The sum of the algebraic remainders of line 1 is equal to the determinant obtained by replacing all elements of line 1 of the original determinant with 1, and the sum of the algebraic remainders of line 2 is equal to the determinant obtained by replacing all elements of line 2 of the original determinant with 1,The sum of the algebraic remainders of the nth row is equal to the determinant obtained by replacing all elements of the nth row of the original determinant with 1. The sum of all algebraic remainders is the sum of the n new determinants above.

Coopter M41, M42....

Algebraic coundant formula a41, a42....

You said a41a41 + a42a42....It's the value of the determinant. The value of the determinant is equal to the value of a row (column) element multiplied by its corresponding algebraic remainder, and then summed.