Iterative method for determinant (linear algebraic formula)

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Iterative method of determinant calculation in linear algebra

Statement and introduction

The iterative method of linear algebraic determinant calculation is to use the step-by-step expansion of determinant to find or summarize n Step sum n-1 rank 、n-2 The relation of order and residual order , Then we can calculate the final result of the whole determinant . For example, it can be

Or vice versa (

), In a word, we can find a derivation formula that evolves step by step . Iterative method is also called recursive method .

Iterative method

Forward iteration

According to the given determinant, you can intuitively find n Step sum n-1 The relation of order , This method is called direct iteration . See the following example for details ：

Calculation n Step determinant ：

#1 Ideas Step1 First observe the characteristics of determinant , And sort out the ideas Step2 If we are right 1 The row will be expanded by determinant 2 term , Which corresponds to

Items and of

It is the same in form or structure , In this way, a cycle is formed, that is, iteration . Step3 according to Step2 The method of n、n-1、n-2… 1 Expand the order to get the final result . #2 Practice Step1： According to section 1 Line to line (0 many , Actually only 2 Elements ) Unfold

The result is ：

Step2： because

yes

, Therefore, it is not difficult to conclude from the above summarized relationship that the final result is ：

Derivation summary

According to the given determinant, we can indirectly find n Step sum n-1 The relation of order , And then gradually reduce the order to get the final result . See the following example for details ：

Calculation n Step determinant

#1 Ideas Step1 First observe the characteristics of determinant , And sort out the ideas Step2 If we are right 1 When a row is expanded according to the determinant algebraic cofactor, it is not difficult to find that n Step sum n-1 Order relationship . Step3 summary Step2 The law in , Finally, write the expression and the final result . #2 Practice Step1： According to section 1 Row to original determinant expansion

The results are as follows

Step2: We have the formula

Make some changes

Because here

,

, therefore

Step3： from Step2 Then we get the relation

The final result of step-by-step reduced order expansion is ：

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