Calculate the determinant a 10, calculate the determinant a

I didn't read the answer, I wrote it myself, it should be right.

Decompose the element x in the nth row and the nth column of the nxn order determinant, x=(x+a)-a, and then decompose the determinant into the sum of the two determinants, respectively: |x a a … a a，-a x a … a a，-a -a x…a a，0 0 0 …0 x+a|and |x a a … a a，-a x a … a a，-a -a x…a a，-a -a -a …-a -a|, the value of the former is: (x+a)d, (where d is the n-1 square matrix, and its law is the same as the determinant given in the question), and the value of the latter is:

a(x-a)^(n-1);

Similarly, decompose x into x=(x-a)+a, and decompose the two determinants into |x a a … a a，-a x a … a a，-a -a x…a a，-a -a -a …-a a|，|x a a … a 0，-a x a … a 0，-a -a x…a 0，-a -a -a …-a x-a|, the former has a value of:

a(x+a) (n-1), the value of the latter is (x-a)d, therefore: (x+a)d-a(x-a) (n-1)=(x-a)d+a(x+a) (n-1), and we get: d=[(x+a) (n-1)-(x-a) (n-1)] 2

Bringing d back yields the value of the determinant as [(x+a) n-(x-a) n] 2

|a|It's just a number, and then take itDeterminantEquivalent to the determinant of a 1x1 matrix, of course equal to itself.

A sequence of number is a set of positive integers.

or a finite subset of it) for the defined domain.

is a sequence of ordered numbers. Each number in the sequence is called an item in the sequence. The number in the first place is called the first term of the series (usually also called the first term), the number in the second place is called the second term of the series, and so on, the number in the nth position of the series is called the nth term of the series, which is usually denoted by an.

The famous number series is the Fibonacci sequence.

Trigonometric functions, Cattelan numbers, Yang Hui triangles, etc.

The term in the sequence must be a number, which can be either a real number or a complex number.

Representing a series of numbers with symbols is nothing more than "borrowing" the symbols of a set, and there are essential differences between them:1The elements in a set are different from each other, while the items in a sequence can be the same.

2.The elements in a set are disordered, while the terms in a sequence must be arranged in a certain order, that is, they must be ordered.

In general, if a series of numbers starts from the second term, and the difference between each term and its preceding silver term is equal to the same constant, the series is called a difference series.

arithmetic sequence), this constant is called the common difference of the series, and the tolerance is usually denoted by the letter d, and the first n terms are denoted by sn. The contour series can be abbreviated as progression).

Find the determinant of matrix a.

a|and inverse matrices.

a (-1), adjoint matrix.

a* =a| a^(-1)；

Because: a -1=a* |a|；

So: a*=|a｜a＾－1；

a×｜＝a｜a＾－1｜＝｜a｜＾n｜a＾－1｜。

aa^-1=1；

So:|a||a^-1|=1；

a^-1|=1/|a|；

a*|=a|^n/|a|=|a|^(n-1)。

Extended information: The determinant of matrix a is also sometimes denoted as |a|。Absolute values and matrix norms.

This notation is also used, which may be confused with the notation of determinants. However, matrix norms are usually represented by double perpendicular lines, and subscripts can be used.

In addition, the absolute value of the matrix is not defined. For this reason, determinants often use vertical line notation (e.g., Clem's rule).

and sub-formulas). For example, there is a determinant |a(i,j)|(i,j is the subscript), if it is now assumed that according to line 1, we know that the Yuanhuai trap in line 1 of the first row is a(1,1),a(1,2),a(1,n), according to line 1, is to multiply the elements in row 1 above by the corresponding cocoon.

Add it up again. i.e. a(1,1)*m(1,1)+a(1,2)*m(1,2)+a(1,n)*m(1,n)。

A matrixDeterminantis a value, and the determinant of a value is himself.

The rows and columns can be seen as Sakurakure is the concept of directed area or volume in general Euclidean space.

or, in n Euclidean space, the determinant describes a linear transformation.

Effect on Volume.

A number is multiplied by a matrix, and then the determinant is taken, then it is equal to the nth power of this number multiplied by the determinant of the original matrix.

Nature

A row (or column) in column and column spring bend 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.

As you can see, the matrix A in the question is a fourth-order matrix.

and then by the adjoint matrix.

basic nature.

aa*=|a|e

Notice that to the right of the equation is a matrix of fourth-order quantities, i.e., its diagonal.

The elements on are all numerous|a|.

The two sides take the ranks and ranks in a vain style.

, the left side is |a||a*|

On the right is the product of the elements on the diagonal, i.e., |a|^4。

The letter is round. a||a*|=a|^4。

aa*=|a|e

This formula should know that it is rotten, so take the determinant on both sides of this formula and get it.

a| |a*| a|e |

And the basal ridge is obvious| |a|e |=a|n, so the edge leaks.

a| |a*| a|^n

Thereupon. a*| a|^ n-1)

a* is the adjoint matrix of a, and there are also textbooks called transposed adjoint matrices.

The element in a* is made up of |a|The algebraic remainder of the elements in is formed.

a* =aji), aij is |a|Algebraic coadopter of aij.

It has the property aa* =a*a = a|e

**Theorem to the determinant.

a|is the determinant of a.

Also detainted, a* refers to the adjoint matrix of matrix a.

is an algebraic coadopter of an element of a.

A sibling matrix formed in the order in which the columns and columns of objects are exchanged.

Definition of adjoint matrix: The algebraic remainder of each element of a matrix a, which is transposed after forming a new matrix, is called the accompaniment of a.

The algebraic remainder of an element is the determinant of the matrix formed by removing the row and column elements of an element in the matrix, and then multiplying it by the power of -1 (number of rows + number of columns).

The differences are as follows: 1. The results are different.

The matrix is a **, and the number of rows and columns can be different; The determinant is a number, and the number of rows must be equal to the number of columns. Only a square matrix can define its determinant, and for a rectangular matrix its determinant cannot be defined. The equality of the two matrices means that the corresponding elements are equal; The equality of two determinants does not require that the corresponding elements are equal, and even the orders can be different, as long as the algebra and result of the operation are the same.

2. The operation method is different.

The addition of two matrices is the addition of each corresponding element; The addition of two determinants is to add the result of the operation, and in special cases (such as having the same row or column), only the elements of one row (or column) can be added, and the rest of the elements are written as they are.

3. The nature is different.

A number multiplication matrix refers to the number multiplied by each element of the matrix; The number multiplication determinant can only be used to multiply a row or column of the determinant to mention the common factor.

And so it is. 4. The result after transformation is different.

The matrix is primarily transformed.

Its rank does not change; The value of the determinant may change after the primary transformation: the change of the method transformation should be changed, and the difference of the multiple transformation should be multiplied; Elimination.

Transformations do not change.

Put the 2nd, 3rd ,..n columns are added to column 1, and the common factor x+(n-1)a is proposed, and :

1 a ..a

1 x ..a

1 a ..x

Multiply -1 on line 1 to add to 2,3 ,..n line, got:

1 a ..a

0 x-a ..0

0 0 ..x-a

It's an upper triangle.

So the determinant = [x+(n-1)a](x-a) (n-1).