Course: 8.7 儿科决定因素

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儿科决定因素

A determinant is a number computed from the entries in a square matrix . It has many properties and interpretations that you will explore in linear algebra. This concept is focused on the procedure of calculating determinants. Once you know how to calculate the determinant of a $\begin{aligned} 2 \times 2 \end{aligned}$ matrix, then you will be able to calculate the determinant of a $\begin{aligned} 3 \times 3 \end{aligned}$ matrix. Once you know how to calculate the determinant of a $\begin{aligned} 3 \times 3 \end{aligned}$ matrix you can calculate the determinant of a $\begin{aligned} 4 \times 4 \end{aligned}$ and so on.
::自动确定从平方矩阵条目中计算的数字。它有许多属性和解释, 您将以线性代数来探索。此概念侧重于计算决定因素的程序。一旦知道如何计算 A2x2 矩阵的决定因素, 您就可以计算 a3x3matrix的决定因素。一旦知道如何计算 A3x3 矩阵的决定因素, 您可以计算 A4x4 的决定因素等等。

A logical question about determinants is where does the procedure come from? Why are determinants defined in the way that they are?
::关于决定因素的一个合乎逻辑的问题是,程序来自何方?为什么确定决定因素的方式是这样的?

The Determinant
::决定因素

The determinant of a matrix $\begin{aligned} A \end{aligned}$ is written as $\begin{aligned} | A | \end{aligned}$ . For a $\begin{aligned} 2 \times 2 \end{aligned}$ matrix $\begin{aligned} A \end{aligned}$ , the value is calculated as:
::矩阵 A 的决定因素以 A 写成。对于 2x2 矩阵 A , 数值计算为 :

\begin{aligned} A & = [\begin{matrix} a & b \\ c & d \end{matrix}] \\ det A & = | A | = | \begin{matrix} a & b \\ c & d \end{matrix} | = a d - b c \end{aligned}

::A=[abcd] did AAAabcdad-bc

If you substitute numbers for the letters and try to calculate $\begin{aligned} det A \end{aligned}$ for $\begin{aligned} A = [\begin{matrix} 3 & 2 \\ 1 & 5 \end{matrix}] \end{aligned}$ , you get:
::如果您用数字替代字母,并试图计算 A=[3215] 的 det A 时,可获得:

$\begin{aligned} | \begin{matrix} 3 & 2 \\ 1 & 5 \end{matrix} | = 3 \cdot 5 - 2 \cdot 1 = 15 - 2 = 13 \end{aligned}$

Notice how the diagonals are multiplied and then subtracted.
::注意对角线如何乘以,然后减去。

The determinant of a $\begin{aligned} 3 \times 3 \end{aligned}$ matrix is more involved.
::3x3矩阵的决定因素涉及较多。

$\begin{aligned} B = [\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}] \end{aligned}$
::B = [abcdefghi]

Usually you will start by looking at the top row, although any row or column will work. Then use the checkerboard pattern for signs (shown below) and create smaller $\begin{aligned} 2 \times 2 \end{aligned}$ matrices.
::通常您会首先查看上行, 尽管任何行或列都会工作。然后使用检查板模式来显示符号( 显示在下面) , 并创建较小的 2x2 矩阵。

$\begin{aligned} [\begin{matrix} + & - & + \\ - & + & - \\ + & - & + \end{matrix}] \end{aligned}$

The smaller $\begin{aligned} 2 \times 2 \end{aligned}$ matrices are the entries that remain when the row and column of the coefficient you are working with are ignored.
::较小的 2x2 矩阵是当您正在使用的系数的行和列被忽略时留下的条目。

\begin{aligned} det B = | B | = + a \cdot | \begin{matrix} e & f \\ h & i \end{matrix} | - b \cdot | \begin{matrix} d & f \\ g & i \end{matrix} | + c \cdot | \begin{matrix} d & e \\ g & h \end{matrix} | \end{aligned}

Next take the determinant of the smaller $\begin{aligned} 2 \times 2 \end{aligned}$ matrices and you get a long string of computations.
::下一步取小于 2x2 矩阵的决定因素,然后进行一长串计算。

\begin{aligned} = + a (e i - f h) - b (d i - f g) + c (d h - e g) \\ = a e i - a f h - b d i + b f g + c d h - c e g \\ = a e i + b f g + c d h - c e g - a f h - b d i \end{aligned}

::*a(ei-fh)-b(di-fg)+c(dh-eg)=aei-afh-bdi+bfg+cdh-ceg=aei+bfg+cdh-ceg-afh-bdi

Most people do not remember this . A French mathematician named Sarrus demonstrated a great device to memorize the computation of the determinant for $\begin{aligned} 3 \times 3 \end{aligned}$ matrices. The first step is simply to copy the first two columns to the right of the matrix. Then draw three diagonal lines going down and to the right.
::大多数人都不记得这一点。一个叫Sarrus的法国数学家展示了一个伟大的装置,可以记住计算3x3 矩阵的决定因素。第一步只是将前两列复制到矩阵右侧。然后绘制三条对角线向下和向右。

$\begin{aligned} B = [\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}] \end{aligned}$
::B = [abcdefghi]

Notice that they correspond exactly to the three positive terms of the determinant demonstrated above. Next draw three diagonals going up and to the right. These diagonals correspond exactly to the three negative terms.
::注意它们完全符合上述决定因素的三个正值条件。下一步绘制三个对角线向右和向右。这些对角线与三个负值条件完全吻合。

\begin{aligned} det B = a e i + b f g + c d h - c e g - a f h - b d i \end{aligned}

::dat B=aei+bfg+cdh-ceg-afh-bdi =aei+bfg+cdh-ceg-afh-bdi =aei+bfg+cdh-ceg-efh-bdi

Sarrus’s Rule does not work for the determinants of matrices that are not of order $\begin{aligned} 3 \times 3 \end{aligned}$ .
::Sarrus的规则不适用于第3x3号命令以外的矩阵的决定因素。

Examples
::实例

Example 1
::例1

Earlier, you were asked where the procedure for finding the determinants came from. Determinants for $\begin{aligned} 2 \times 2 \end{aligned}$ matrices are defined the way they are because of the general solution to a system of 2 variables and 2 equations.
::早些时候,有人问您查找决定因素的程序来自何方。 2x2 矩阵的决定因素被定义为由于对2个变量和2个方程的系统的一般解决方案而形成的方式。

\begin{aligned} a x + b y & = e \\ c x + d y & = f \end{aligned}

::ax+by=ecx+dy=f =x+by=ecx+dy=f =xx=ecx+by=ecx+dy=f = ax+by=ecx+dy=f

To eliminate the $\begin{aligned} x \end{aligned}$ , scale the first equation by $\begin{aligned} c \end{aligned}$ and the second equation by a.
::要删除 x, 将第一个方程式缩放为 c, 第二个方程式缩放为 a 。

\begin{aligned} a c x + b c y & = e c \\ a c x + a d y & = a f \end{aligned}

::acx+bcy=ecacx+ady=af 缩写

Subtract the second equation from the first and solve for $\begin{aligned} y \end{aligned}$ .
::将第二个方程从第一个方程中减去,然后解决y。

\begin{aligned} a d y - b c y & = a f - e c \\ y (a d - b c) & = a f - e c \\ y & = \frac{a f - e c}{a d - b c} \end{aligned}

::a-bcy=af-ecy(ad-bc)=af-ecy=af-ecad-bc

When you solve for $\begin{aligned} x \end{aligned}$ you also get $\begin{aligned} a d - b c \end{aligned}$ in the denominator of the general solution. This pattern led people to start using this strategy in solving systems of equations. The determinant is defined in this way so it will always be the denominator of the general solution of either variable .
::当对 x 进行解析时,您也会在通用解决方案的分母中获得 ad-bc 。这种模式导致人们开始使用此策略来解析方程式系统。以这种方式定义决定因素, 以便它始终是任一变量的一般解决方案的分母。

Example 2
::例2

Find the determinant of the following matrix.
::找出下列矩阵的决定因素。

$\begin{aligned} C = [\begin{matrix} - 4 & 12 \\ 1 & - 3 \end{matrix}] \end{aligned}$
::C=[-4121-3]

\begin{aligned} det C = | \begin{matrix} - 4 & 12 \\ 1 & - 3 \end{matrix} | = 12 - 12 = 0 \end{aligned}

::C*4121 -3*12 -12=0

Example 3
::例3

Find $\begin{aligned} det B \end{aligned}$ for $\begin{aligned} B = [\begin{matrix} 3 & 2 & 1 \\ 5 & 0 & 2 \\ 2 & 1 & 5 \end{matrix}] \end{aligned}$
::B=[321502215]的查找 det B

$\begin{aligned} | \begin{matrix} 3 & 2 & 1 \\ 5 & 0 & 2 \\ 2 & 1 & 5 \end{matrix} | & = 3 | \begin{matrix} 0 & 2 \\ 1 & 5 \end{matrix} | - 2 | \begin{matrix} 5 & 2 \\ 2 & 5 \end{matrix} | + 1 | \begin{matrix} 5 & 0 \\ 2 & 1 \end{matrix} | \\ = 3 (0 \cdot 5 - 2 \cdot 1) - 2 (5 \cdot 5 - 2 \cdot 2) + 1 (5 \cdot 1 - 2 \cdot 0) \\ = - 6 - 42 + 5 = - 43 \end{aligned}$

Example 4
::例4

Find the determinant of $\begin{aligned} B \end{aligned}$ from example B using Sarrus’s Rule.
::使用Sarrus规则从例B中找出B的决定因素。

$\begin{aligned} \begin{matrix} 3 & 2 & 1 & 3 & 2 \\ 5 & 0 & 2 & 5 & 0 \\ 2 & 1 & 5 & 2 & 1 \end{matrix} \end{aligned}$

$\begin{aligned} det B = 0 + 8 + 5 - 0 - 6 - 50 = - 43 \end{aligned}$

::B=0+8+5-5-0-6-50{_________________________________________________________________________________________________________________________________________________________________________________________________________________________________

As you can see, Sarrus’s Rule is efficient and much of the calculations can be done mentally. Additionally, zero values make much of the multiplication easier.
::如你所见,Sarrus的规则是有效的,许多计算可以在精神上进行。此外,零值让乘法更容易得多。

Example 5
::例5

Find the determinant of the following $\begin{aligned} 4 \times 4 \end{aligned}$ matrix by carefully choosing the row or column to work with.
::仔细选择要工作的行或列,以查找以下 4x4 矩阵的决定因素。

$\begin{aligned} E = [\begin{matrix} 4 & 5 & 0 & 2 \\ - 1 & - 3 & 0 & 3 \\ 4 & 8 & 1 & 5 \\ - 3 & 2 & 0 & 9 \end{matrix}] \end{aligned}$
::E=[4502-1-3034815-3209]

Notice that the third column is made up with zeros and a one. Choose this column to make up the coefficients because then instead of having to evaluate the determinant of four individual $\begin{aligned} 3 \times 3 \end{aligned}$ matrices, you only need to do one.
::请注意, 第三列由零和一组成。选择此列来组成系数, 因为这样您不必评估 4 个单个 3x3 矩阵的决定因素, 您只需要做一个。

\begin{aligned} | \begin{matrix} 4 & 5 & 0 & 2 \\ - 1 & - 3 & 0 & 3 \\ 4 & 8 & 1 & 5 \\ - 3 & 2 & 0 & 9 \end{matrix} | & = 0 \cdot | \begin{matrix} - 1 & - 3 & 3 \\ 4 & 8 & 5 \\ - 3 & 2 & 9 \end{matrix} | - 0 \cdot | \begin{matrix} 4 & 5 & 2 \\ 4 & 8 & 5 \\ - 3 & 2 & 9 \end{matrix} | + 1 \cdot | \begin{matrix} 4 & 5 & 2 \\ - 1 & - 3 & 3 \\ - 3 & 2 & 9 \end{matrix} | - 0 \cdot | \begin{matrix} 4 & 5 & 2 \\ - 1 & - 3 & 3 \\ 4 & 8 & 5 \end{matrix} | \\ = | \begin{matrix} 4 & 5 & 2 \\ - 1 & - 3 & 3 \\ - 3 & 2 & 9 \end{matrix} | \\ = 4 \cdot (- 3) \cdot 9 + 5 \cdot 3 \cdot (- 3) + 2 \cdot (- 1) \cdot 2 - 18 - 24 - (- 45) \\ = - 154 \end{aligned}

Summary

A determinant is a number computed from the entries in a square matrix.
::决定因素是从平方矩阵条目中计算的数字。

The determinant of a 2x2 matrix is calculated as: $\begin{aligned} | A | = a d - b c . \end{aligned}$
::2x2矩阵的决定因素计算如下:

For a 3x3 matrix, the determinant can be calculated using the checkerboard pattern for signs and creating smaller 2x2 matrices.
::对于 3x3 矩阵, 决定因素可以使用标记的检查板模式计算, 并创建较小的 2x2 矩阵。

Sarrus's Rule is a helpful device for memorizing the computation of the determinant for 3x3 matrices, involving copying the first two columns and drawing diagonal lines.
::Sarrus的规则是一个有用的工具,用于计算3x3矩阵的决定因素,包括复制前两栏和绘制对角线。

Sarrus's Rule does not work for determinants of matrices that are not of order 3x3.
::Sarrus的规则不适用于非顺序3x3的矩阵的决定因素。

Review
::回顾

Find the determinants of each of the following matrices.
::找出下列各矩阵的决定因素。

1. $\begin{aligned} [\begin{matrix} 4 & 5 \\ 2 & 3 \end{matrix}] \end{aligned}$

2. $\begin{aligned} [\begin{matrix} - 3 & 6 \\ 2 & 5 \end{matrix}] \end{aligned}$

3. $\begin{aligned} [\begin{matrix} - 1 & 2 \\ 2 & 0 \end{matrix}] \end{aligned}$

4. $\begin{aligned} [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \end{aligned}$

5. $\begin{aligned} [\begin{matrix} 6 & 5 \\ 2 & - 2 \end{matrix}] \end{aligned}$

6. $\begin{aligned} [\begin{matrix} 1 & 2 \\ 6 & 3 \end{matrix}] \end{aligned}$

7. $\begin{aligned} [\begin{matrix} - 1 & 3 & - 4 \\ 4 & 2 & 1 \\ 1 & 2 & 5 \end{matrix}] \end{aligned}$

8. $\begin{aligned} [\begin{matrix} 4 & 5 & 8 \\ 9 & 0 & 1 \\ 0 & 3 & - 2 \end{matrix}] \end{aligned}$

9. $\begin{aligned} [\begin{matrix} 0 & 7 & - 1 \\ 2 & - 3 & 1 \\ 6 & 8 & 0 \end{matrix}] \end{aligned}$

10. $\begin{aligned} [\begin{matrix} 4 & 2 & - 3 \\ 2 & 4 & 5 \\ 1 & 8 & 0 \end{matrix}] \end{aligned}$

11. $\begin{aligned} [\begin{matrix} - 2 & - 6 & - 12 \\ - 1 & - 5 & - 2 \\ 2 & 3 & 4 \end{matrix}] \end{aligned}$

12. $\begin{aligned} [\begin{matrix} - 2 & 6 & 3 \\ 2 & 4 & 0 \\ - 8 & 2 & 1 \end{matrix}] \end{aligned}$

13. $\begin{aligned} [\begin{matrix} 2 & 6 & 4 & 6 \\ 0 & 1 & 0 & 1 \\ 2 & 4 & 2 & 0 \\ - 6 & 2 & 3 & 1 \end{matrix}] \end{aligned}$

14. $\begin{aligned} [\begin{matrix} 5 & 0 & 0 & 1 \\ 2 & 1 & 8 & 3 \\ 9 & 3 & 2 & 6 \\ - 4 & 2 & 5 & 1 \end{matrix}] \end{aligned}$

15. Can you find the determinant for any matrix? Explain.
::15. 你能找到任何矩阵的决定因素吗?

16. The following matrix has a determinant of zero: $\begin{aligned} [\begin{matrix} 6 & 4 \\ 3 & 2 \end{matrix}] \end{aligned}$ . If the determinant of a matrix is zero, what does that say about the rows of the matrix?
::16. 下列矩阵有一个零的决定因素:[6432],如果矩阵的决定因素是零,那么矩阵的行数又如何?

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Section outline

General

儿科决定因素

The Determinant ::决定因素

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)