Section: 以 Scalar 乘以 Scalar 的矩阵 | 4.3 乘以Scalar的乘数母体

Section outline

A group of 8 kids, a group of 8 adults, and a group of 8 seniors are attending a movie.
::一组8个孩子,一组8个成年人, 和一组8个老人正在看电影。

For a matinee movie, the movie theater charges the following prices:
::电影院售价如下:

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 determine the total amount each group will be charged for each type of movie?
::我们如何确定每一类电影的收费总额?

Multiplying Matrices by a Scalar
::以 Scalar 乘以 Scalar 的矩阵

A matrix can be multiplied by a scalar. A scalar is a real number in matrix algebra-not a matrix. To multiply a matrix by a scalar, each element in the matrix is multiplied by the scalar as shown below:
::矩阵可乘以弧度。弧度在矩阵代数中是一个实际数字, 代数不是矩阵。要将矩阵乘以弧度, 矩阵中的每个元素乘以弧度, 如下所示:

$\begin{aligned} k [\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} k a & k b \\ k c & k d \end{matrix}], where k is a scalar . \end{aligned}$

::k [abcd] = [kakbkkkkkd], k 是一个 calar 。

Distributive Property of Scalar Multiplication
::Scal 乘法分配属性

Let’s investigate what happens if we distribute the multiplication of a scalar of the addition of two matrices. Consider the matrix expression :
$\begin{aligned} 3 ([\begin{matrix} 2 \\ - 5 \end{matrix}] + [\begin{matrix} - 3 \\ 6 \end{matrix}]) \end{aligned}$

::让我们来调查如果我们分配增加两个矩阵的斜体乘法的乘数, 会发生什么。考虑矩阵表达式 : 3( 3( 2 - 5) + [ - 36])

Method 1: Perform the addition inside the parenthesis first and then multiply by the scalar:
::方法1:首先在括号内添加,然后乘以弧度:

$\begin{aligned} 3 ([\begin{matrix} 2 \\ - 5 \end{matrix}] + [\begin{matrix} - 3 \\ 6 \end{matrix}]) = 3 [\begin{matrix} - 1 \\ 1 \end{matrix}] = [\begin{matrix} - 3 \\ 3 \end{matrix}] \end{aligned}$

Method 2: Distribute the scalar into both matrices and then add:
::方法2:在两个矩阵中分配标量,然后添加:

$\begin{aligned} 3 ([\begin{matrix} 2 \\ - 5 \end{matrix}] + [\begin{matrix} - 3 \\ 6 \end{matrix}]) = [\begin{matrix} 6 \\ - 15 \end{matrix}] + [\begin{matrix} - 9 \\ 18 \end{matrix}] = [\begin{matrix} - 3 \\ 3 \end{matrix}] \end{aligned}$

The results are equivalent. We can conjecture that the of Multiplication over Addition is true for scalar multiplication of matrices. This property can be extended to include distribution of scalar multiplication over subtraction as well.
::结果相等。我们可以推断乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法乘法

Distributive Property of Addition: $\begin{aligned} k (A + B) = k A + k B \end{aligned}$
::添加:k(A+B)=kA+kB的分配属性

Distributive Property of Subtraction: $\begin{aligned} k (A - B) = k A - K B \end{aligned}$
::减法分配属性:k(A-B)=kA-KB

Let's perform the scalar multiplication for the matrix below:
$\begin{aligned} 2 [\begin{matrix} - 4 & \frac{1}{2} \\ - 1 & 3 \end{matrix}] \end{aligned}$

::让我们为下面的矩阵执行天平乘法: 2[- 412- 13]

In this case we just need to multiply each element of the matrix by 2.
::在这种情况下,我们只需要将矩阵的每个要素乘以2即可。

$\begin{aligned} [\begin{matrix} 2 (- 4) & 2 (\frac{1}{2}) \\ 2 (- 1) & 2 (3) \end{matrix}] = [\begin{matrix} - 8 & 1 \\ - 2 & 6 \end{matrix}] \end{aligned}$

Perform the indicated operations for the matrix below.
::为下面的矩阵执行指定的操作。

$\begin{aligned} 3 ([\begin{matrix} 2 \\ - 1 \end{matrix}] + [\begin{matrix} \frac{2}{3} \\ 4 \end{matrix}]) \end{aligned}$

This time we need to decide whether to distribute the 3 inside the parenthesis or add first and then multiply by 3. It is possible to complete this problem either way. However, careful observation allows us to observe that there is a fraction inside the second matrix. By distributing the 3 first, we can eliminate this fraction and make the addition easier as shown below.
::这一次我们需要决定是否在括号内分配3个, 或者首先添加3个, 然后乘以3 。但是, 仔细观察让我们可以观察到第二个矩阵内有一小部分。通过分配第3个矩阵, 我们可以消除这一部分, 并方便添加如下。

$\begin{aligned} (3 [\begin{matrix} 2 \\ - 1 \end{matrix}] + 3 [\begin{matrix} \frac{2}{3} \\ 4 \end{matrix}]) = ([\begin{matrix} 6 \\ - 3 \end{matrix}] + [\begin{matrix} 2 \\ 12 \end{matrix}]) = [\begin{matrix} 8 \\ 9 \end{matrix}] \end{aligned}$

Perform the indicated operations for the matrix below.
::为下面的矩阵执行指定的操作。

$\begin{aligned} \frac{1}{2} ([\begin{matrix} 7 & - 1 \\ 2 & 8 \end{matrix}] + [\begin{matrix} - 3 & 5 \\ 2 & 0 \end{matrix}]) \end{aligned}$

Again, we need to decide whether to do the multiplication or addition first. Here, it turns out to be easier to add first and then multiply as shown below.
::同样,我们需要决定是首先进行乘法还是增加法。在这里,首先增加比较容易,然后再增加,如下文所示。

$\begin{aligned} \frac{1}{2} [\begin{matrix} 7 + - 3 & - 1 + 5 \\ 2 + 2 & 8 + 0 \end{matrix}] = \frac{1}{2} [\begin{matrix} 4 & 4 \\ 4 & 8 \end{matrix}] = [\begin{matrix} 2 & 2 \\ 2 & 4 \end{matrix}] \end{aligned}$

Not that this problem could have been solved in the other order , but we would have had to deal with fractions.
::这并不是说这个问题本来可以按其他顺序解决,但我们不得不处理分数问题。

Examples
::实例

Example 1
::例1

Earlier, you were asked to determine the total amount each group will be charged for each type of movie.
::早些时候,你被要求确定每一类电影将收取的费用总额。

We could organize the data in a matrix and multiply it by the scalar 8.
::我们可以将数据组织在一个矩阵中乘以8号天梯

$\begin{aligned} K A S \\ \begin{matrix} Matinee \\ Night \end{matrix} & 8 [\begin{matrix} 5 & 8 & 6 \\ 7 & 10 & 8 \end{matrix}] = \end{aligned}$

$\begin{aligned} K A S \\ \begin{matrix} Matinee \\ Night \end{matrix} & [\begin{matrix} 40 & 64 & 48 \\ 56 & 80 & 64 \end{matrix}] \end{aligned}$

::[406448568064] [5867108] = Kasmatiane夜 [406448568064]

We can now easily see that the group of adults will be charged $64 for a matinee, the group of kids will be charged $56 at night, etc.
::现在我们很容易看出, 成人组将收取64美元, 一组儿童将收取56美元, 晚上等等。

Example 2
::例2

Perform the indicated operation .
::执行指示的操作。

$\begin{aligned} \frac{2}{3} [\begin{matrix} 0 \\ 6 \\ 9 \end{matrix}] \end{aligned}$

Multiply each element inside the matrix by $\begin{aligned} \frac{2}{3} \end{aligned}$ :
::将矩阵内的每个元素乘以 23 :

$\begin{aligned} [\begin{matrix} \frac{2}{3} (0) \\ \frac{2}{3} (6) \\ \frac{2}{3} (9) \end{matrix}] = [\begin{matrix} 0 \\ 4 \\ 6 \end{matrix}] \end{aligned}$

Example 3
::例3

Perform the indicated operation.
::执行指示的操作。

$\begin{aligned} - \frac{2}{3} ([\begin{matrix} 2 & - 3 & 5 \end{matrix}] - [\begin{matrix} - 1 & 0 & 2 \end{matrix}]) \end{aligned}$

If we subtract what is inside the parenthesis first, we can avoid fractions:
::如果我们先减去括号内的内容, 我们可以避免分数 :

$\begin{aligned} - \frac{2}{3} ([\begin{matrix} 2 - (- 1) & - 3 - 0 & 5 - 2 \end{matrix}]) = - \frac{2}{3} [\begin{matrix} 3 & - 3 & 3 \end{matrix}] = [\begin{matrix} - 2 & 2 & - 2 \end{matrix}] \end{aligned}$

Example 4
::例4

Perform the indicated operation.
::执行指示的操作。

$\begin{aligned} 12 ([\begin{matrix} \frac{3}{4} & - \frac{2}{3} \\ \frac{1}{6} & 2 \end{matrix}] + [\begin{matrix} 1 & \frac{5}{6} \\ \frac{2}{3} & \frac{5}{4} \end{matrix}]) \end{aligned}$

If we distribute first this time, we can avoid fractions:
::如果我们第一次分发, 我们可以避免分数 :

$\begin{aligned} (12 [\begin{matrix} \frac{3}{4} & - \frac{2}{3} \\ \frac{1}{6} & 2 \end{matrix}] + 12 [\begin{matrix} 1 & \frac{5}{6} \\ \frac{2}{3} & \frac{5}{4} \end{matrix}]) = [\begin{matrix} 9 & - 8 \\ 2 & 24 \end{matrix}] + [\begin{matrix} 12 & 10 \\ 8 & 15 \end{matrix}] = [\begin{matrix} 21 & 2 \\ 10 & 39 \end{matrix}] \end{aligned}$

Review
::回顾

Perform the indicated operations, if possible.
::在可能的情况下,执行指示的操作。

.

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

.

$\begin{aligned} - 2 [\begin{matrix} - 6 & 8 \\ 5 & - 2 \end{matrix}] \end{aligned}$

.

$\begin{aligned} \frac{2}{3} [\begin{matrix} 12 \\ 6 \end{matrix}] \end{aligned}$

.

$\begin{aligned} - \frac{3}{2} [\begin{matrix} 8 & 0 & - 4 \\ - 6 & 2 & 10 \end{matrix}] \end{aligned}$

.

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

.

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

.

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

.

$\begin{aligned} 3 [\begin{matrix} 4 & - 5 & - 1 \end{matrix}] + 4 [\begin{matrix} 8 & - 1 & 5 \end{matrix}] \end{aligned}$

.

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

.

$\begin{aligned} - 2 ([\begin{matrix} \frac{1}{2} & \frac{5}{2} \end{matrix}] - [\begin{matrix} 3 & - 1 \end{matrix}]) \end{aligned}$

.

$\begin{aligned} - \frac{1}{3} ([\begin{matrix} 5 \\ - 2 \end{matrix}] - [\begin{matrix} 2 \\ 4 \end{matrix}]) \end{aligned}$

.

$\begin{aligned} 3 [\begin{matrix} 8 & - 2 \\ \frac{1}{3} & 5 \end{matrix}] + [\begin{matrix} 2 & 4 \\ - 6 & - 1 \end{matrix}] \end{aligned}$

.

$\begin{aligned} 2 [\begin{matrix} \frac{3}{7} \\ \frac{3}{4} \end{matrix}] \end{aligned}$

.

$\begin{aligned} - 6 ([\begin{matrix} \frac{1}{3} & 0 \\ 2 & - \frac{2}{3} \end{matrix}] + [\begin{matrix} 5 & 1 \\ - \frac{1}{6} & 2 \end{matrix}]) \end{aligned}$

.

$\begin{aligned} [\begin{matrix} - 2 \\ 3 \\ 8 \end{matrix}] - 5 ([\begin{matrix} \frac{1}{5} \\ - \frac{2}{5} \\ \frac{11}{5} \end{matrix}] + [\begin{matrix} - 2 \\ \frac{1}{5} \\ - \frac{3}{5} \end{matrix}]) \end{aligned}$

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

Multiplying Matrices by a Scalar ::以 Scalar 乘以 Scalar 的矩阵

Distributive Property of Scalar Multiplication ::Scal 乘法分配属性

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)

Multiplying Matrices by a Scalar
::以 Scalar 乘以 Scalar 的矩阵

Distributive Property of Scalar Multiplication
::Scal 乘法分配属性

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)