Course: 4.1 界定和比较矩阵

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界定和比较矩阵
For a matinee movie, a movie theater charges the following prices:
::电影院收取以下价格:

Kids: $5 Adults: $8 Seniors: $6
::儿童:5美元成人:8美元老年人:6美元

For the same movie at night, the theater charges the following prices:
::在同一部电影的夜间, 剧院收取以下价格:

Kids: $7 Adults: $10 Seniors: $8
::儿童:7美元成人:10美元老年人:8美元

How could we organize this data to easily compare the prices?
::我们如何组织这种数据来方便地比较价格?

Matrices
::儿数

A matrix consists of data that is organized into rows and columns to form a rectangle. For example, we could organize the data collected at a movie theater concession stand during a matinee show into the follow matrix:
::矩阵由按行和列排列的数据组成,形成矩形。例如,我们可以将电影院特许展台收集的数据组织起来,然后在以下矩阵中展示:

$\begin{aligned} S M L \\ \begin{matrix} popcorn \\ soda \end{matrix} & [\begin{matrix} 20 & 46 & 32 \\ 15 & 53 & 29 \end{matrix}] \end{aligned}$
$\begin{align*}& \quad S \quad M \quad L\\ \begin{matrix} \text{popcorn}\\ \quad \ \text{soda}\end{matrix} & \begin{bmatrix} 20 & 46 & 32\\ 15 & 53 & 29\end{bmatrix}\end{align*}$
::苏打汽水[204632155329]

Now we can easily compare the quantities of each size sold. These values in the matrix are called elements.
::现在我们可以很容易地比较每个售出大小的数量。矩阵中的这些值被称为元素。

This particular matrix has two rows and three columns. Matrices are often described in terms of dimensions (rows by columns). This matrix is a $\begin{align*}2 \times 3\end{align*}$ (read as '2 by 3') matrix.
::这个特定的矩阵有两行和三列,矩阵通常用尺寸(逐列)来描述。这个矩阵是一个 2x3 矩阵(读作“2乘3”) 矩阵。

The variables $\begin{align*}m\end{align*}$ (rows) and $\begin{align*}n\end{align*}$ (columns) are most often used to represent unknown dimensions. Matrices in which the number of rows equals the number of columns $\begin{align*}(m=n)\end{align*}$ are called square matrices .
::变量 m (行) 和 n (列) 通常用于代表未知的维度。行数等于列数( m=n) 的矩阵称为平方矩阵。

Matrices which have the same dimensions and all corresponding elements equal are said to be equal matrices .
::具有相同维度和所有相应元素相等的母体据说是平等的矩阵。

Let's solve the following problems about matrices.
::让我们解决以下关于矩阵的问题。

Using the matrix above, what is the value of the element in the second row, second column?
::使用上面的矩阵,第二行第二列中元素值是多少?

$\begin{aligned} Column 2 \\ ↓ \\ S M L \\ \begin{matrix} popcorn \\ Row 2 \to soda \end{matrix} [\begin{matrix} 20 & 46 & 32 \\ 15 & 53 & 29 \end{matrix}] \end{aligned}$
$\begin{align*}& \qquad \qquad \qquad \quad {\color{red}{\text{Column 2}}}\\ & \qquad \qquad \quad \quad \qquad \ \ \downarrow\\ & \quad \qquad \qquad \ \ \ \quad S \quad \ \ M \ \ \quad L\\ & \begin{matrix} \qquad \ \ \text{popcorn}\\ {\color{red}{\text{Row 2}}} \rightarrow \text{soda}\end{matrix} \begin{bmatrix} 20 & 46 & 32\\ 15 & \boxed{53} & 29 \end{bmatrix}\end{align*}$
::第2列 S M L 爆米花Row 2soda[204632155329]

We must see where the second row and second column overlap and identify the element in that location. In this case. it is 53.
::我们必须看到第二行和第二列的重叠之处,并确定该位置的元素。在这种情况下,是53。

Determine the dimensions $\begin{align*}(m \times n)\end{align*}$ of the matrices below.
::确定以下矩阵的尺寸(mxn)。

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

This matrix has 2 rows and 2 columns. Therefore it is a $\begin{align*}2 \times 2\end{align*}$ matrix.
::这个矩阵有两个行和两个列。因此它是一个 2x2 矩阵。

$\begin{aligned} [\begin{matrix} 4 & - 3 & 2 & 7 \\ 3 & 5 & - 4 & 6 \\ 9 & 1 & 0 & - 2 \end{matrix}] \end{aligned}$
$\begin{align*}\begin{bmatrix} 4 & -3 & 2 & 7\\ 3 & 5 & -4 & 6\\ 9 & 1 & 0 & -2 \end{bmatrix}\end{align*}$

This matrix has 3 rows and 4 columns. Therefore it is a $\begin{align*}3 \times 4\end{align*}$ matrix.
::这个矩阵有3行和4列,因此是一个 3x4 矩阵。

$\begin{aligned} [\begin{matrix} 2 \\ - 3 \\ 1 \end{matrix}] \end{aligned}$
$\begin{align*}\begin{bmatrix} 2\\ -3\\ 1 \end{bmatrix}\end{align*}$

This matrix has 3 rows and 1 column. Therefore it is a $\begin{align*}3 \times 1\end{align*}$ matrix.
::该矩阵有3行和1列,因此是一个 3x1 矩阵。

Which two matrices are equal? Explain your answer.
::哪些两个矩阵相等? 请解释您的答复。

$\begin{aligned} A = [\begin{matrix} 1 & - 5 \\ - 2 & 4 \\ 8 & 3 \end{matrix}] B = [\begin{matrix} - 5 & 4 & 3 \\ 1 & - 2 & 8 \end{matrix}] C = [\begin{matrix} 1 & - 5 \\ - 2 & 4 \\ 8 & 3 \end{matrix}] \end{aligned}$
$\begin{align*}A = \begin{bmatrix} 1 & -5\\ -2 & 4\\ 8 & 3 \end{bmatrix} \qquad B = \begin{bmatrix} -5 & 4 & 3\\ 1 & -2 & 8 \end{bmatrix} \qquad C = \begin{bmatrix} 1 & -5\\ -2 & 4\\ 8 & 3 \end{bmatrix}\end{align*}$
::A=[1-5-2483]B=[-543-128]C=[1-5-2483]

Matrices $\begin{align*}A\end{align*}$ and $\begin{align*}C\end{align*}$ are equal matrices. They are both $\begin{align*}3 \times 2\end{align*}$ matrices and have all of the same elements. Matrix $\begin{align*}B\end{align*}$ is a $\begin{align*}2 \times 3\end{align*}$ matrix so even though it contains the same elements, they are arranged differently preventing it from being equal to the other two.
::矩阵 A 和 C 是等式矩阵。它们都是 3x2 矩阵, 具有所有相同的元素。矩阵 B 是一个 2x3 矩阵, 即使包含相同的元素 , 它们的排列也不同 , 无法等同其它两个元素。

Examples
::实例

Example 1
::例1

Earlier, you were asked how to organize the data to easily compare the prices of a movie theater.
::早些时候,你被问及如何组织数据以方便地比较电影院的价格。

To make it easy to compare prices, we could organize the data in matrix like this one:
::为了便于比较价格,我们可以将数据组织成像这样的矩阵:

$\begin{aligned} K A S \\ \begin{matrix} Matinee \\ Night \end{matrix} & [\begin{matrix} 5 & 8 & 6 \\ 7 & 10 & 8 \end{matrix}] \end{aligned}$
$\begin{align*} & \ \ \ K \quad A \quad S\\ \begin{matrix} \text{Matinee}\\ \ \ \ \text{Night}\end{matrix} & \begin{bmatrix} 5 & 8 & 6\\ 7 & 10 & 8\end{bmatrix}\end{align*}$
::[5867108]

Example 2
::例2

What are the dimensions of the matrix: $\begin{align*}[ 3 \quad -5 \quad 1 \quad 0]\end{align*}$ ?
::矩阵的层面是什么:[3-510]?

The dimensions are $\begin{align*}1 \times 4\end{align*}$ .
::尺寸是1×4

Example 3
::例3

In the matrix
$\begin{aligned} [\begin{matrix} 8 & - 5 & 4 \\ - 2 & 6 & - 3 \\ 3 & 0 & - 7 \\ 1 & 3 & 9 \end{matrix}] \end{aligned}$
$\begin{align*}\begin{bmatrix} 8 & -5 & 4\\ -2 & 6 & -3\\ 3 & 0 & -7\\ 1 & 3 & 9 \end{bmatrix}\end{align*}$ what is the element in the second row, third column?
::在矩阵[8-54-26-330-7139]中,第二行第三列的元素是什么?

The element in the second row, third column is -3 as shown below:
::第二行中的元素,第三列为 -3,如下所示:

$\begin{aligned} Column 3 \\ ↓ \\ \begin{matrix} Row 2 \to \end{matrix} [\begin{matrix} 8 & - 5 & 4 \\ - 2 & 6 & - 3 \\ 3 & 0 & - 7 \\ 1 & 3 & 9 \end{matrix}] \end{aligned}$
$\begin{align*}& \qquad \qquad \qquad \qquad {\color{red}{\text{Column 3}}}\\ & \qquad \quad \qquad \qquad \qquad \ \downarrow\\ & \begin{matrix} {\color{red}{\text{Row 2 }}} \rightarrow \end{matrix} \begin{bmatrix} \\ 8 & -5 & 4\\ -2 & 6 & \boxed{-3}\\ 3 & 0 & -7\\ 1 & 3 & 9\end{bmatrix}\end{align*}$
::第3列列 2 [8-54-26-330-7139]

Example 4
::例4

Are the matrices $\begin{align*}A = [-1 \quad 4 \quad 9]\end{align*}$ and
$\begin{aligned} B = [\begin{matrix} - 1 \\ 4 \\ 9 \end{matrix}] \end{aligned}$
$\begin{align*}B = \begin{bmatrix} -1\\ 4\\ 9 \end{bmatrix}\end{align*}$ equal matrices?
::矩阵A=[-149]和B=[-149]是否相等?

No, $\begin{align*}A\end{align*}$ and $\begin{align*}B\end{align*}$ are not equal matrices. They have the same elements, but the dimensions are not the same.
::不,A和B不是平等的矩阵。它们有相同的元素,但尺寸不同。

Review
::回顾

Use the matrices below to answer questions 1-7 that follow:
::使用以下矩阵回答以下问题1-7:

$\begin{aligned} A = [\begin{matrix} 2 & 3 & 1 \\ - 5 & - 8 & 4 \end{matrix}] B = [\begin{matrix} 2 & 1 \\ - 3 & 5 \end{matrix}] C = [\begin{matrix} - 5 & 1 & 3 \\ 8 & - 2 & 6 \\ 4 & 9 & 7 \end{matrix}] \end{aligned}$
$\begin{align*}A = \begin{bmatrix} 2 & 3 & 1\\ -5 & -8 & 4 \end{bmatrix} \qquad B = \begin{bmatrix} 2 & 1\\ -3 & 5 \end{bmatrix} \qquad C = \begin{bmatrix} -5 & 1 & 3\\ 8 & -2 & 6\\ 4 & 9 & 7 \end{bmatrix}\end{align*}$
::A=[231-5-84]B=[21-35]C=[-5138-26497]

$\begin{aligned} D = [\begin{matrix} 2 & 1 \\ - 3 & 5 \end{matrix}] E = [\begin{matrix} - 5 & 2 \\ - 8 & 3 \\ 4 & 1 \end{matrix}] F = [\begin{matrix} 5 & - 1 & 8 \\ - 2 & 6 & - 3 \end{matrix}] \end{aligned}$
$\begin{align*}D = \begin{bmatrix} 2 & 1\\ -3 & 5 \end{bmatrix} \qquad E = \begin{bmatrix} -5 & 2\\ -8 & 3\\ 4 & 1 \end{bmatrix} \qquad F = \begin{bmatrix} 5 & -1 & 8\\ -2 & 6 & -3\\ \end{bmatrix}\end{align*}$
::D=[21-35]E=[-52-52-8341]F=[5-18-26-3]

What are the dimensions of

Matrix $\begin{align*}B\end{align*}$ ?
::矩阵B?

Matrix $\begin{align*}E\end{align*}$ ?
::E矩阵?

Matrix $\begin{align*}F\end{align*}$ ?
::F矩阵?

::母体B的维度是多少?母体E的维度是多少?母体F的维度是多少?母体F的维度是多少?

Which matrices have the same dimensions?
::哪些矩阵具有相同的维度?

Which matrices are square matrices?
::哪些矩阵是方矩阵?

Which matrices are equal?
::哪些矩阵是相等的?

What is the element in row 1, column 2 of Matrix $\begin{align*}C\end{align*}$ ?
::矩阵C第1行第2列的元素是什么?

What is the element in row 3, column 1 of Matrix $\begin{align*}E\end{align*}$ ?
::矩阵E第1列第3行的元素是什么?

What is the element in row 1, column 1 of Matrix $\begin{align*}D\end{align*}$ ?
::矩阵D第1列第1行的元素是什么?

Write a matrix with dimensions $\begin{align*}3 \times 4\end{align*}$ .
::以维度 3x4 写一个矩阵。

Write a matrix with dimensions $\begin{align*}7 \times 2\end{align*}$ .
::写一个尺寸为 7x2 的矩阵。

For problems 10-14, determine if the statements are true or false.
::对于10-14的问题,请确定这些声明是真实的还是虚假的。

A $\begin{align*}3 \times 2\end{align*}$ and a $\begin{align*}2 \times 3\end{align*}$ are equal.
::A 3x2 和 a 2x3 等值。

Two matrices are equal if every element within the two matrices is the same.
::如果两个矩阵中的每个要素相同,则两个矩阵相同。

A matrix is a way to organize data.
::矩阵是组织数据的一种方法。

The element in row 2, column 2 in $\begin{align*}F\end{align*}$ above is -1.
::第2行,F栏2中的元素是-1。

The element in row 2, column 2 in $\begin{align*}F\end{align*}$ above is 6.
::第2行2栏F栏2的元素是6。

Organize the data into a matrix: A math teacher gave her class three tests during the semester. On the first test there were 10 A’s, 8 B’s, 12 C’s, 4 D’s and 1 F. On the second test there were 8 A’s, 11 B’s, 14 C’s, 2 D’s and 0 F’s. On the third test there were 13 A’s, 7 B’s, 8 C’s, 4 D’s and 3 F’s.
::将数据组织成一个矩阵:一位数学教师在学期给了她三次考试。在第一个测试中,有10个A、8个B、12个C、4个D和1个F。在第二个测试中,有8个A、11个B、14个C、2个D和0个F。在第三个测试中,有13个A、7个B、8个C、4个D和3个F。

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

界定和比较矩阵

Matrices ::儿数

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Matrices
::儿数

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)