Review of Linear Algebra: determinant

1. determinant ： It's a value

To find a determinant is to find the value of this determinant
Two , Third order determinant ： It can be used ： Diagonal rule and sand method do
Diagonal rule ： Subtract... From the sum of the main diagonals Sum of sub diagonal products .
a b
c d : The value is ad-bc

Be careful ：n rank ：n That's ok n Column .

2. If it is n How to find the order determinant ？

1. Lower triangle rule （ Above the main diagonal are 0）：
The determinant is transformed into the product of the value of the lower triangular determinant equal to the value of the element of the main diagonal .
Like the upper triangle .

2. Is the determinant expansion ：
Keep turning large determinants into small determinants ：
determinant = Elements * Algebraic cofactor （ This line is not... Except for one element 0, Everything else is 0, It's equal to the value of this element multiplied by the algebraic cofactor of this element ）

The remainder formula ： Cross out the row and column of an element ： Get a new determinant ： This determinant is the cofactor of this element

Algebraic cofactor ： Namely （-1） Of (i+j) Power * The remainder formula

Note here ： Symbol
M: Denotes the remainder （ With subscript ）
A: Represents the algebraic cofactor
The element of a determinant is ：a
The sign of determinant ：D

3. The nature of determinants ：

1. A number multiplied by a determinant is equal to the number multiplied by the determinant of each element of a row or column of the determinant
2. The determinant remains unchanged ： Calculate the determinant obtained by multiplying one row or column by a number and adding it to another row or column ： It's the symbol
r1+r2k
c1+c2k

Problem practice 1：

To calculate the determinant ： Determinant expansion