Eye of depth (III) -- determinant of matrix

One 、 The definition of determinant

1.1 Second order determinant

Second order determinant ： Multiply the main diagonal - Multiply by sub diagonal

1.2 Third order determinant

For the time being, no matter how it is calculated
D2 If so, replace the second column with b₁、b₂、b₃,D3 Is to replace the third column with b₁、b₂、b₃,

Two 、 Calculation of determinant

2.1 Full Permutation and inverse sequence

2.2 Calculation definition

First of all, yes n*n The matrix of is n The whole arrangement , The sign is determined by the inverse ordinal number , The number in reverse order is odd , It's negative , The number in reverse order is even , It's positive , It can be popularized in turn

3、 ... and 、 Determinant of special matrix and properties of determinant

3.1 Determinants of special matrices

According to this , Observe an item , You know , Take one from the first row and multiply it by the second row to get a different column from the first row , By analogy , Form a summation .


therefore , For the first picture （ Principal diagonal matrix ）, Although there are n The whole arrangement , But there will be no satisfaction 0 One of （ Because there is 0 Words , This item is 0 了 ）, And meet different columns , So only the main diagonal is multiplied , The signs are in normal order , So the number in reverse order is 0, So it's a plus sign

The second picture （ Negative diagonal matrix ）（ The one below the first picture ） Only the symbol problem , Just do the calculation

The third picture , Lower triangular matrix
The first row will only take the first column （ Because other columns are 0, Multiply to 0, No influence ）, The second row will only take the second column , And so on , It's the same as the first picture

3.2 The nature of determinants

This is the case. , Push it yourself , Mainly understand the essence , These properties are only abstractions of essence , Don't forget

Four 、 Determinant by line （ Column ） an , Algebraic cofactor

Reduced order processing , Use the lower order determinant to calculate the higher order determinant

Is to use the idea of a special matrix , Only one in a row or column is not 0, Then for that row, only that column is valid , And that column has different principles according to the rest of the lines , So don't look , Therefore, we get the above conclusion

Theorem 3

inference

5、 ... and 、 Application of determinant in linear equations ： Clem's law

Clem's law （ Idealization ）： The unknown number is the same as the number of equations , And the coefficient determinant of the equation is not equal to 0, Then the equation has a unique solution

D_j Remove the corresponding column , Switch to b Constant column of

Left nonhomogeneous , Right homogeneous