课程： 10.5 矩阵代数 | The Way To Learn

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矩阵代数
Introduction
::导言

The local movie theater is running a special for the Martin Luther King Jr. holiday weekend. All matinee tickets are $5, regardless of the age of the customer.
::当地电影院为小马丁路德金节周末举办特别节目,

The following matrix shows the number of each type of matinee ticket the movie theater sold over the Martin Luther King Jr. holiday weekend:
::以下矩阵显示了在小马丁路德金节周末出售的电影院每张类型的门票的数量:

Tickets Sold
::售票

$\begin{aligned} Sat Sun Mon \\ \begin{matrix} Kids \\ Adults \\ Seniors \end{matrix} 5 \cdot & [\begin{matrix} 122 & 133 & 150 \\ 89 & 75 & 101 \\ 57 & 38 & 49 \end{matrix}] \end{aligned}$

::SatSunMon儿童成人老人5[1221331508975101573849]

How much total money did the movie theater take in for the three-day weekend from kids' ticket sales?
::电影院花了多少钱在三天的周末从孩子的票销售?

Matrix Algebra
::矩阵代数

Algebra refers to the ability to manipulate variables and unknowns based on rules and properties. Matrix algebra is extremely similar to the algebra you already know for numbers, with a few important differences.
::代数指根据规则和属性操作变量和未知数的能力。矩阵代数与您已经知道的数字代数非常相似,有一些重要的差异。

Two matrices of the same order can be added by summing the entries in the corresponding positions:
::以相应的位置对条目进行汇总,可以添加相同顺序的两个矩阵:

$\begin{aligned} [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}] + [\begin{matrix} 6 & 5 & 4 \\ 3 & 2 & 1 \end{matrix}] = [\begin{matrix} 7 & 7 & 7 \\ 7 & 7 & 7 \end{matrix}] \end{aligned}$

Two matrices of the same order can be subtracted by subtracting the entries in the corresponding positions:
::可以通过减去相应职位中的条目来减去同一顺序的两个矩阵:

$\begin{aligned} [\begin{matrix} 10 & 9 & 8 \\ 7 & 6 & 5 \end{matrix}] - [\begin{matrix} 2 & 2 & 2 \\ 2 & 2 & 2 \end{matrix}] = [\begin{matrix} 8 & 7 & 6 \\ 5 & 4 & 3 \end{matrix}] \end{aligned}$

The following video further demonstrates how to add, subtract, and perform scalar multiplication with matrices:
::以下视频进一步展示了如何与矩阵增加、减法和进行天平乘法的乘法:

You can find the product of matrix $\begin{aligned} A \end{aligned}$ and matrix $\begin{aligned} B \end{aligned}$ if the number of columns in matrix $\begin{aligned} A \end{aligned}$ matches the number of rows in matrix $\begin{aligned} B \end{aligned}$ . Another way to remember this is to write the order (number of rows x number of columns) of matrix $\begin{aligned} A \end{aligned}$ and the order (number of rows x number of columns) of matrix $\begin{aligned} B \end{aligned}$ next to each other. You can find the product of the matrices if the inner numbers are the same. The matrix that results from multiplying matrix A and matrix B will have the same number of rows as matrix A, and the same number of columns as matrix B .
::如果矩阵A和矩阵B的列数与矩阵B的行数相符,您可以找到矩阵A和矩阵B的产物。另一个记住的方法是将矩阵A的顺序(行数x列数)和矩阵B的顺序(行数x列数)相邻写入。如果内数相同,您可以找到矩阵的产物。乘进矩阵A和矩阵B的矩阵与矩阵A的行数相同,列数与矩阵B相同。

$\begin{aligned} (2 \times 3) \cdot (3 \times 5) = (2 \times 5) \end{aligned}$

To compute the 1st entry of the resulting $\begin{aligned} 2 \times 5 \end{aligned}$ matrix, you should match the 1st row from the 1st matrix and the 1st column of the 2nd matrix. The arithmetic operation to combine these numbers is identical to taking the dot product between two vectors.
::要计算结果 2x5 矩阵的第一个条目, 您应该匹配第一个矩阵的第一行和第二个矩阵的第一列。合并这些数值的算术操作与在两个矢量之间取出点产品相同。

The entry in the 1st row, 1st column of the new matrix is computed as $\begin{aligned} 1 \cdot 0 + 4 \cdot 2 + 3 \cdot 1 = 11 \end{aligned}$ .
::第1行,新矩阵第1列的条目以1+0+4+2+31=11计算。

The entry in the 2nd row, 1st column of the new matrix is computed as $\begin{aligned} 5 \cdot 0 + 6 \cdot 2 + 9 \cdot 1 = 21 \end{aligned}$ .
::第2行,新矩阵第1列条目的计算为 50+62+91=21。

The rest of the entries of this product are left to Example 1, below.
::该产品的其他条目留待下文例1处理。

Properties of Matrix Algebra
::矩阵代数属性

The commutative property holds for matrix addition . This means that when matrices $\begin{aligned} A \end{aligned}$ and $\begin{aligned} B \end{aligned}$ can be added (when they have matching orders), then $\begin{aligned} A + B = B + A \end{aligned}$ .
::通量属性保留在矩阵中,这意味着当可以添加矩阵A和B(当它们有匹配的订单时),然后是A+B=B+A。

The commutative property does not hold in general for . For this reason, we need to be careful to distribute without changing the order of the resulting multiplication.
::因此,我们必须谨慎分配,不改变由此引起的乘法的顺序。

The associative property does hold for both multiplication and addition. $\begin{aligned} (A B) C = A (B C), (A + B) + C = A + (B + C) \end{aligned}$ .
::连带财产确实持有乘法和加法。 (AB)C=A(BC),(A+B)+C=A(B+C)

Distribution over addition and subtraction holds. $\begin{aligned} A (B \pm C) = A B \pm A C \end{aligned}$ .
::A(BC)=ABAC。

The following video further explains how to multiply matrices:
::以下影片进一步解释如何乘法矩阵:

The following video demonstrates how to multiply matrices on the TI 83/84:
::以下影片展示了如何将TI 83/84上的矩阵乘以:

Play, Learn, and Explore Matrix Algebra: .
::游戏、学习和探索矩阵代数 :

Examples
::实例

Example 1
::例1

Complete the entries of the matrix multiplication introduced in the guidance section, above.
::完成上文指导一节中引入的矩阵乘法条目。

$\begin{aligned} [\begin{matrix} 1 & 4 & 3 \\ 5 & 6 & 9 \end{matrix}] \cdot [\begin{matrix} 0 & 1 & 3 & 1 & 0 \\ 2 & 0 & 0 & 2 & 1 \\ 1 & 1 & 3 & 0 & 1 \end{matrix}] \end{aligned}$

Solution:
::解决方案 :

Two of the arithmetic operations are shown here:
::此处显示两个算术操作:

$\begin{aligned} c_{12} & = 1 \cdot 1 + 4 \cdot 0 + 3 \cdot 1 = 4 \\ c_{22} & = 5 \cdot 1 + 6 \cdot 0 + 9 \cdot 1 = 14 \end{aligned}$

::c12=11+40+31=4c22=51+60+91=14

When the rest are completed, the result is as follows:
::其余部分完成后,结果如下:

$\begin{aligned} C & = [\begin{matrix} 11 & 4 & 12 & 9 & 7 \\ 21 & 14 & 42 & 17 & 15 \end{matrix}] \end{aligned}$

::C=[11412972114421715]

Example 2
::例2

Show the commutative property does not hold by demonstrating $\begin{aligned} A B \neq B A \end{aligned}$ for:
::显示通量属性时不会通过显示 ABBA 来显示 :

$\begin{aligned} A = [\begin{matrix} 0 & - 1 & 8 \\ 1 & 2 & 0 \\ 4 & 3 & 12 \end{matrix}], B = [\begin{matrix} 1 & 5 & 1 \\ 2 & 2 & 1 \\ 4 & 3 & 0 \end{matrix}] \end{aligned}$

::A=[0-181204312],B=[151221430]

Solution:
::解决方案 :

$\begin{aligned} A B & = [\begin{matrix} 30 & 22 & - 1 \\ 5 & 9 & 3 \\ 58 & 62 & 7 \end{matrix}] \\ B A & = [\begin{matrix} 9 & 12 & 20 \\ 6 & 5 & 28 \\ 3 & 2 & 32 \end{matrix}] \end{aligned}$

::AB=[3022-159358627]BA=[9122065283232]

Example 3
::例3

Compute the following matrix arithmetic, $\begin{aligned} 10 \cdot (2 A - 3 C) \cdot B, \end{aligned}$ for the following matrices:
::计算下列矩阵算术10(2A-3C)B,用于下列矩阵:

$\begin{aligned} A = [\begin{matrix} 1 & 2 \\ 4 & 5 \end{matrix}], B = [\begin{matrix} 0 & 1 & 2 \\ 4 & 3 & 2 \end{matrix}], C = [\begin{matrix} 12 & 0 \\ 1 & 3 \end{matrix}] \end{aligned}$

::A=[1245],B=[012432],C=[2013]

Solution:
::解决方案 :

When a matrix is multiplied by a scalar (such as with $\begin{aligned} 2 A \end{aligned}$ ), multiply each entry in the matrix by the scalar.
::当一个矩阵乘以一个标量(如2A)时,将矩阵中每个条目乘以标量。

$\begin{aligned} 2 A & = [\begin{matrix} 2 & 4 \\ 8 & 10 \end{matrix}] \\ - 3 C & = [\begin{matrix} - 36 & 0 \\ - 3 & - 9 \end{matrix}] \\ 2 A - 3 C & = [\begin{matrix} - 34 & 4 \\ 5 & 1 \end{matrix}] \end{aligned}$

::2A=[24810]-3C=[-360-3-3-9]2A-3C=[-344451]

Since the associative property holds, you can either distribute the 10 or multiply by matrix $\begin{aligned} B \end{aligned}$ next.
::自关联财产持有以来,您可以分配 10 或乘以矩阵 B 。

$\begin{aligned} (2 A - 3 C) \cdot B & = [\begin{matrix} 16 & - 22 & - 60 \\ 4 & 8 & 12 \end{matrix}] \\ 10 \cdot (2 A - 3 C) \cdot B & = [\begin{matrix} 160 & - 220 & - 600 \\ 40 & 80 & 120 \end{matrix}] \end{aligned}$

:2A-3C)B=[16-22-604812]10(2A-3C)B=[160-220-6004080120]

Example 4
::例4

Recall the question from the Introduction: W hat is the total amount of money the movie theater took in from kids' ticket sales for the three-day weekend?
::回顾导言中的问题:电影院从儿童3天周末的票销售中拿到的钱总额是多少?

Solution:
::解决方案 :

This is a matrix multiplication problem where the matrix is multiplied by a scalar.
::这是一个矩阵乘法问题,当矩阵乘以一个卡路里时。

Tickets Sold
::售票

$\begin{aligned} Sat Sun Mon \\ \begin{matrix} Kids \\ Adults \\ Seniors \end{matrix} 5 \cdot & [\begin{matrix} 122 & 133 & 150 \\ 89 & 75 & 101 \\ 57 & 38 & 49 \end{matrix}] \end{aligned}$

::SatSunMon儿童成人老人5[1221331508975101573849]

Use your calculator to assist you. The resulting matrix is the following:
::使用计算器来帮助您。由此产生的矩阵如下:

Tickets Sales ($)
$\begin{aligned} Sat Sun Mon \\ \begin{matrix} Kids \\ Adults \\ Seniors \end{matrix} [\begin{matrix} 610 & 665 & 750 \\ 445 & 375 & 505 \\ 285 & 190 & 245 \end{matrix}] \end{aligned}$

::销售车票 (美元) 圣善蒙幼年成年人[61066575045457575505285190245]

From this matrix, we can see that the movie theater made $610 from kids' ticket sales on Saturday, $665 on Sunday, and $750 on Monday. Therefore , the total amount made from kids' ticket sales was $\begin{aligned} $ 610 + $ 665 + $ 750 = $ 2, 025. \end{aligned}$
::从这个矩阵中,我们可以看到,电影院在星期六、星期天和星期一的票价销售中分别从儿童票价销售中拿出了610美元、665美元和750美元。因此,儿童票销售总额是610美元+665美元+750美元=2,025美元。

Example 5
::例5

Show that a $\begin{aligned} 3 \times 3 \end{aligned}$ identity matrix works as the multiplicative identity .
::显示一个 3x3 身份矩阵作为倍复制身份有效。

Solution:
::解决方案 :

A $\begin{aligned} 3 \times 3 \end{aligned}$ matrix multiplied by the identity should yield the original matrix.
::A 3x3 矩阵乘以身份,应得出原始矩阵。

$\begin{aligned} [\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}] \cdot [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}] \\ a_{11} = a \cdot 1 + b \cdot 0 + c \cdot 0 = a \\ a_{12} = a \cdot 0 + b \cdot 1 + c \cdot 0 = b \\ ⋮ \end{aligned}$

::[abcdefghi][1001001001]=[abcdefghi]a11=a1+b0+c0=a12=a0+b1+c0=b

Example 6
::例6

Use your calculator to input and compute the following matrix operations:
::使用计算器输入和计算下列矩阵操作:

$\begin{aligned} A = [\begin{matrix} 54 & 65 & 12 \\ 235 & 322 & 167 \\ 413 & 512 & 123 \end{matrix}], B = [\begin{matrix} 163 & 212 & 466 \\ 91 & 221 & 184 \\ 42 & 55 & 42 \end{matrix}] \\ A^{T} \cdot B \cdot A - 100 A \end{aligned}$

::A=[5465122335322167413512123],B=[6321246691221184425542]ATBA-100A

Solution:
::解决方案 :

Most graphing calculators, such as the TI-84, can do operations on matrices. Find where you can enter matrices and enter the following two matrices:
::大多数图形计算器,如TI-84,可以在矩阵上操作。请查找您在哪里可以输入矩阵并输入以下两个矩阵:

Then type in the appropriate operation and see the result. The TI-84 has a built-in Transpose button.
::然后在适当的操作中键入并查看结果。 TI- 84 中有一个内嵌的转换按钮。

The actual numbers on this practice are less important than knowing that your calculator can perform all the matrix algebra demonstrated in this concept. It is useful to be aware of the capabilities of the tools at your disposal, but this recognition should not replace knowing why the calculator does what it does.
::这种做法的实际数字比知道您的计算器能够执行这个概念所显示的所有矩阵代数要不重要,了解你所掌握的工具的能力是有益的,但这种认识不应取代知道计算器为什么做它的工作。

Example 7
::例7

Matrix multiplication can be used as a transformation in the coordinate system. Consider the triangle with coordinates (0, 0), (1, 2), and (1, 0), and the following matrix:
::矩阵乘法可用作坐标系统中的转换。考虑带有坐标(0,0,0,1,2)和(1,0)的三角形以及下列矩阵:

$\begin{aligned} [\begin{matrix} \cos 90^{\circ} & \sin 90^{\circ} \\ - \sin 90 & \cos 90 \end{matrix}] \end{aligned}$

::[cos909090990909090]

What does the new picture look like?
::新照片长什么样?

Solution:
::解决方案 :

The matrix simplifies to become:
::矩阵简化为:

$\begin{aligned} [\begin{matrix} \cos 90^{\circ} & \sin 90^{\circ} \\ - \sin 90 & \cos 90 \end{matrix}] = [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}] \end{aligned}$

::[cos9090909090909090]=[01~10]

When applied to each point as a transformation, a new point is produced. Note that $\begin{aligned} [\begin{matrix} x & y \end{matrix}] \end{aligned}$ is a matrix representing each original point, and $\begin{aligned} [\begin{matrix} x^{'} & y^{'} \end{matrix}] \end{aligned}$ is the new point. The $\begin{aligned} x^{'} \end{aligned}$ is read as " $\begin{aligned} x \end{aligned}$ prime," and is a common way to refer to a result after a transformation.
::当应用到作为转换的每个点时,将产生一个新点。注意 [xy] 是代表每个原始点的矩阵, [x_ y_ ] 是新点。 x_ 被解读为“ x print ” , 并且是一种常见的表示转换后结果的方法。

$\begin{aligned} [\begin{matrix} x & y \end{matrix}] \cdot [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}] & = [\begin{matrix} x^{'} & y^{'} \end{matrix}] \\ [\begin{matrix} 0 & 0 \end{matrix}] \cdot [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}] & = [\begin{matrix} 0 & 0 \end{matrix}] \\ [\begin{matrix} 1 & 2 \end{matrix}] \cdot [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}] & = [\begin{matrix} - 2 & 1 \end{matrix}] \\ [\begin{matrix} 1 & 0 \end{matrix}] \cdot [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}] & = [\begin{matrix} 0 & 1 \end{matrix}] \end{aligned}$

::[xy]_[01-10]=[x_Y_[00]_[01-10]=[00][12]_[01-10]=[-21][10]_[01-10]_[01-10]=[01]=[01]

Notice how the matrix transformation rotates graphs in a counterclockwise direction $\begin{aligned} 90^{\circ} \end{aligned}$ .
::注意矩阵转换如何在逆时针方向90 旋转图表。

$\begin{aligned} [\begin{matrix} x & y \end{matrix}] [\begin{matrix} \cos 90^{\circ} & \sin 90^{\circ} \\ - \sin 90 & \cos 90 \end{matrix}] = [\begin{matrix} - y & x \end{matrix}] \end{aligned}$

::[xy]=[-yx]

The matrix transformation applied in the order below will rotate a graph clockwise $\begin{aligned} 90^{\circ} \end{aligned}$ .
::按以下顺序应用的矩阵变换将按时钟顺时针90旋转一个图表。

$\begin{aligned} [\begin{matrix} \cos 90^{\circ} & \sin 90^{\circ} \\ - \sin 90 & \cos 90 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} y \\ - x \end{matrix}] \end{aligned}$

::[cos 90 90 90 90 [xy] =[y-x]

Summary
::摘要

Matrix operations are addition, subtraction, and multiplication. Division involves a multiplicative inverse that we will learn about in a future section .
::矩阵操作是附加、减法和乘法。分区涉及一个多倍反向,我们将在今后的一节中了解。

The commutative property holds for matrix addition. $\begin{aligned} A + B = B + A \end{aligned}$ .
::A+B=B+A。

The commutative property does not hold in general for matrix multiplication.
::通量财产一般不用于矩阵乘法。

The associative property holds for both multiplication and addition. $\begin{aligned} (A B) C = A (B C), (A + B) + C = A + (B + C) \end{aligned}$ .
::连带财产同时持有乘数和加数。 (AB)C=A(BC),(A+B)+C=A(B+C)

Distribution over addition and subtraction holds. $\begin{aligned} A (B \pm C) = A B \pm A C \end{aligned}$ .
::A(BC)=ABAC。

Review
::回顾

Attempt numbers 1-11 without your calculator.
::没有计算器就尝试了1-11号

$\begin{aligned} A = [\begin{matrix} 2 & 7 \\ 3 & 8 \end{matrix}], B = [\begin{matrix} 0 & 5 & 1 \\ 3 & 4 & 6 \end{matrix}], C = [\begin{matrix} 14 & 6 \\ 1 & 2 \end{matrix}], D = [\begin{matrix} 5 & 0 \\ 1 & 2 \end{matrix}] \end{aligned}$
::A=[2738],B=[051346],C=[14612],D=[5012]

1. Find $\begin{aligned} A C \end{aligned}$ . If not possible, explain.
::1. 寻找AC。如果不可能,请解释。

2. Find $\begin{aligned} B A \end{aligned}$ . If not possible, explain.
::2. 寻找BA。如果不可能,请解释。

3. Find $\begin{aligned} C A \end{aligned}$ . If not possible, explain.
::3. 寻找CA。如果不可能,请解释。

4. Find $\begin{aligned} 4 B^{T} . \end{aligned}$ If not possible, explain.
::4. 查找4BT。如果不可能,请解释。

5. Find $\begin{aligned} A + C . \end{aligned}$ If not possible, explain.
::5. 查找A+C。如果不可能,请解释。

6. Find $\begin{aligned} D - A \end{aligned}$ . If not possible, explain.
::6. 寻找D-A. 如果不可能,请解释。

7. Find $\begin{aligned} 2 (A + C - D) \end{aligned}$ . If not possible, explain.
::7. 查找2(A+C-D)。如果不可能,请解释。

8. Find $\begin{aligned} (A + C) B . \end{aligned}$ If not possible, explain.
::8. 查找(A+C)B. 如果不可能,请解释。

9. Find $\begin{aligned} B (A + C) \end{aligned}$ . If not possible, explain.
::9. 查找B(A+C)。如果不可能,请解释。

10. Show that $\begin{aligned} A (C + D) = A C + A D . \end{aligned}$
::10. 显示A(C+D)=AC+AD。

11. Show that $\begin{aligned} A (C - D) = A C - A D . \end{aligned}$
::11. 显示A(C-D)=AC-AD。

Practice using your calculator for 12-15.
::12 -15使用计算器练习

$\begin{aligned} E = [\begin{matrix} 312 & 59 & 34 \\ 342 & 156 & 189 \\ 783 & 23 & 133 \end{matrix}], F = [\begin{matrix} 33 & 72 & 21 \\ 93 & 41 & 94 \\ 62 & 75 & 72 \end{matrix}], G = [\begin{matrix} 11 & 735 & 67 \\ 93 & 456 & 2 \\ 94 & 34 & 0 \end{matrix}] \end{aligned}$
::E=[312593434215618978323133],F=[337221934194627572],G=[1173556793456294340]

12. Find $\begin{aligned} E + F + G \end{aligned}$ .
::12. 寻找E+F+G。

13. Find $\begin{aligned} 2 E \end{aligned}$ .
::13. 查找2E。

14. Find $\begin{aligned} 4 F \end{aligned}$ .
::14. 发现4F。

15. Find $\begin{aligned} (E + F) G \end{aligned}$ .
::15. 查找(E+F)G.

16. What are the differences between algebra and matrix algebra?
::16. 代数和矩阵代数之间有什么区别?

17. A movie theater tracks the sale of popcorn in three different sizes. The data collected over a weekend (Friday, Saturday, and Sunday nights) is shown in the matrix below. The price of each size is shown in a 2nd matrix. How much revenue did the theater take in each night for popcorn? How much did the theater take in for popcorn in total?
::17. 电影院追踪爆米花销售的三个不同大小:周末(星期五、星期六和星期天晚上)收集的数据见下面的矩阵;每个大小的价格见第二个矩阵;每晚剧院的爆米花收入有多少?这些剧院总共花了多少?

$\begin{aligned} S M L Price \\ \begin{matrix} Friday \ \\ Saturday \ \\ Sunday \ \end{matrix} & [\begin{matrix} 36 & 85 & 40 \\ 41 & 112 & 51 \\ 28 & 72 & 35 \end{matrix}] \begin{matrix} S \\ M \\ L \end{matrix} [\begin{matrix} 5.50 \\ 6.25 \\ 7.25 \end{matrix}] \end{aligned}$

::[3685404111251287235] SML [5.506.257.25]

Review (Answers)
::回顾(答复)

Please see the Appendix.
::请参看附录。

章节大纲

常规

矩阵代数

Introduction ::导言

Matrix Algebra ::矩阵代数

Examples ::实例

Summary ::摘要

Review ::回顾

Review (Answers) ::回顾(答复)

Introduction
::导言

Matrix Algebra
::矩阵代数

Examples
::实例

Summary
::摘要

Review
::回顾

Review (Answers)
::回顾(答复)