4.2 加法和减法

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  添加和减法

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  F or a matinee movie, a movie theater charges the following prices:
  ::电影院收取以下价格:

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

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  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:

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

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  H ow could we determine how much more the theater charges at night for each ticket type?
  ::我们怎样才能确定每张票 晚上的剧院收费 还要多多少?

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  Adding and Subtracting Matrices
  ::添加和减法

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  If two matrices have the same dimensions , then they can be added or subtracted by adding or subtracting corresponding elements as shown below.
  ::如果两个矩阵的尺寸相同,则可以通过增加或减去下列相应要素而增减。

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  Addition :
  ::加:

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  $$\begin{array}{r}\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]+\left[\begin{array}{cc}e& f\\ g& h\end{array}\right]=\left[\begin{array}{cc}a+e& b+f\\ c+g& d+h\end{array}\right]\end{array}$$

  
  ::[abcd]+[efgh]=[a+eb+fc+gd+h]

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  Subtraction :
  ::减法 :

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  $$\begin{array}{r}\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]-\left[\begin{array}{cc}e& f\\ g& h\end{array}\right]=\left[\begin{array}{cc}a-e& b-f\\ c-g& d-h\end{array}\right]\end{array}$$

  
  ::[abcd]-[efgh]=[a-eb-fc-gd-h]

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  It is important to note that the two matrices are not required to be square matrices . The requirement is that they are the same dimensions. In other words, you can add two matrices that are both $\begin{array}{r}2\times 3\end{array}$ , but you cannot add a $\begin{array}{r}2\times 2\end{array}$ matrix with a $\begin{array}{r}3\times 2\end{array}$ matrix. Before attempting to add two matrices, check to make sure that they have the same dimensions.
  ::必须指出的是,这两个矩阵不需要是平方矩阵。 要求是它们是相同的维度。 换句话说, 您可以添加两个矩阵, 两者都是 2x3, 但是您不能用 3x2 矩阵添加 2x2 矩阵。 在试图添加两个矩阵之前, 请检查以确保它们具有相同的维度 。

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  Commutative and Associative Properties of Addition
  ::增加的动产和共同财产

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  The Commutative Property of Addition states that $\begin{array}{r}a+b=b+a\end{array}$ for real numbers, $\begin{array}{r}a\end{array}$ and $\begin{array}{r}b\end{array}$ . Does this property hold for matrices? The Associative Property of Addition states that $\begin{array}{r}a+(b+c)=(a+b)+c\end{array}$ for real numbers, $\begin{array}{r}a,b\end{array}$ and $\begin{array}{r}c\end{array}$ . Does this property hold for matrices? Consider the matrices below:
  ::添加商品的公用财产规定,a+b=b+a为实际数字,a和b。该财产是否持有矩阵?附加商品的联营财产规定,a+(b+c)=(a+b)+c为实际数字,a、b和c。该财产是否持有矩阵?考虑以下矩阵:

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  $$\begin{array}{r}A=\left[\begin{array}{cc}-3& 7\\ 4& -1\end{array}\right]B=\left[\begin{array}{cc}5& 1\\ -8& -2\end{array}\right]C=\left[\begin{array}{cc}-6& -10\\ 5& 3\end{array}\right]\end{array}$$

  
  ::A=[-374--1]B=[51-8-2]C=[-6-1053]

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  1. Find

  $$\begin{array}{r}A+B\Rightarrow \left[\begin{array}{cc}-3& 7\\ 4& -1\end{array}\right]+\left[\begin{array}{cc}5& 1\\ -8& -2\end{array}\right]=\left[\begin{array}{cc}2& 8\\ -4& -3\end{array}\right]\end{array}$$

  
  ::1. 查找A+B[-374-1]+[51-8-2]=[28-4-3]

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  2. Find

  $$\begin{array}{r}B+A\Rightarrow \left[\begin{array}{cc}5& 1\\ -8& -2\end{array}\right]+\left[\begin{array}{cc}-3& 7\\ 4& -1\end{array}\right]=\left[\begin{array}{cc}2& 8\\ -4& -3\end{array}\right]\end{array}$$

  
  ::2. 查找B+A[51-8-2]+[-374-1]=[28-4-3]

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  Since $\begin{array}{r}A+B=B+A\end{array}$ , we can conjecture that matrix addition is commutative.
  ::自A+B=B+A以来,我们可以推测,增加的矩阵是通融的。

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  3. Find

  $$\begin{array}{r}(A+B)+C\Rightarrow \left(\left[\begin{array}{cc}-3& 7\\ 4& -1\end{array}\right]+\left[\begin{array}{cc}5& 1\\ -8& -2\end{array}\right]\right)+\left[\begin{array}{cc}-6& -10\\ 5& 3\end{array}\right]=\left[\begin{array}{cc}2& 8\\ -4& -3\end{array}\right]+\left[\begin{array}{cc}-6& -10\\ 5& 3\end{array}\right]=\left[\begin{array}{cc}-4& -2\\ 1& 0\end{array}\right]\end{array}$$

  
  ::3. 查找(A+B)+C([-374-1]+[51-8-2])+[-6-1053]=[28-4-3]+[-6-1053]=[-4-210]]

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  4. Find

  $$\begin{array}{r}A+(B+C)\Rightarrow \left[\begin{array}{cc}-3& 7\\ 4& -1\end{array}\right]+\left(\left[\begin{array}{cc}5& 1\\ -8& -2\end{array}\right]+\left[\begin{array}{cc}-6& -10\\ 5& 3\end{array}\right]\right)=\left[\begin{array}{cc}-3& 7\\ 4& -1\end{array}\right]+\left[\begin{array}{cc}-1& -9\\ -3& 1\end{array}\right]=\left[\begin{array}{cc}-4& -2\\ 1& 0\end{array}\right]\end{array}$$

  
  ::4. 查找A+(B+C)[-374-1]+([51-8-2]+[-6-1053])=[-374-1]+[-1-9-31]=[-4-210]]

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  Since $\begin{array}{r}(A+B)+C=A+(B+C)\end{array}$ , we can conjecture that the associative property is true for matrix addition as well.
  ::由于(A+B)+C=A+(B+C),我们可以推断组合属性对于矩阵添加也是真实的。

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  Commutative Property: $\begin{array}{r}A+B=B+A\end{array}$
  ::动产:A+B=B+A

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  Associative Property: $\begin{array}{r}(A+B)+C=A+(B+C)\end{array}$
  :A+B)+C=A+(B+C)

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  $\begin{array}{r}{}^{\ast }\end{array}$ Note that these properties do not work with subtraction with real numbers. For example: $\begin{array}{r}7-5\ne 5-7\end{array}$ . Because they do not hold for subtraction of real numbers, they also do not work with matrix subtraction.
  ::* 注意这些特性与实际数字的减法不起作用,例如:7-55-7。由于这些特性不等于实际数字的减法,因此也与矩阵减法不起作用。

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  Let's find the sum of the following matrices:

  $$\begin{array}{r}\left[\begin{array}{ccc}4& -5& 6\\ -3& 7& 9\end{array}\right]+\left[\begin{array}{ccc}-1& 4& 8\\ 0& -3& 12\end{array}\right]=\end{array}$$

  
  ::我们来看看以下矩阵的总和:[4-56-379]+[-1480-312]=

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  By adding the elements in corresponding positions we get:
  ::通过在相应的立场中增加以下内容:

  $$\begin{array}{r}\left[\begin{array}{ccc}4& -5& 6\\ -3& 7& 9\end{array}\right]+\left[\begin{array}{ccc}-1& 4& 8\\ 0& -3& 12\end{array}\right]=\left[\begin{array}{ccc}4+-1& -5+4& 6+8\\ -3+0& 7+-3& 9+12\end{array}\right]=\left[\begin{array}{ccc}3& -1& 14\\ -3& 4& 21\end{array}\right]\end{array}$$

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  Let's find the difference of the following matrices:

  $$\begin{array}{r}\left[\begin{array}{c}-7\\ 6\\ -9\\ 10\end{array}\right]-\left[\begin{array}{c}-3\\ -2\\ 8\\ 15\end{array}\right]=\end{array}$$

  
  ::让我们发现以下矩阵的差别:[-76-910]-[-3-2815]=

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  By subtracting the elements in corresponding positions we get:
  ::通过在相应的位置上减去元素,我们得到了:

  $$\begin{array}{r}\left[\begin{array}{c}-7\\ 6\\ -9\\ 10\end{array}\right]-\left[\begin{array}{c}-3\\ -2\\ 8\\ 15\end{array}\right]=\left[\begin{array}{c}-7-(-3)\\ 6-(-2)\\ -9-8\\ 10-15\end{array}\right]=\left[\begin{array}{c}-4\\ 8\\ -17\\ -5\end{array}\right]\end{array}$$

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  Now, let's perform the indicated operation :

  $$\begin{array}{r}\left[\begin{array}{cc}-4& 2\\ 5& -3\\ 13& 8\end{array}\right]+\left[\begin{array}{ccc}7& -1& 0\\ -5& 2& 6\end{array}\right]\end{array}$$

  
  ::现在,让我们执行指示的操作 : [ -425 - 3138] + [7 - 10 - 526]

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  In this case the first matrix is $\begin{array}{r}3\times 2\end{array}$ and the second matrix is $\begin{array}{r}2\times 3\end{array}$ . Because they have different dimensions they cannot be added.
  ::在这种情况下,第一个矩阵是3×2,第二个矩阵是2×3,因为它们有不同的维度,因此不能添加。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked to determine how much more the theater charges at night for each ticket. 
  ::早些时候,你被要求确定 每张票晚上的剧场费用 有多少?

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  We could organize the data into two separate matrices and subtract.
  ::我们可以将数据分为两个不同的矩阵和减法。

  $$\begin{array}{r}\left[\begin{array}{ccc}7& 10& 8\end{array}\right]-\left[\begin{array}{ccc}5& 8& 6\end{array}\right]=\left[\begin{array}{ccc}2& 2& 2\end{array}\right]\end{array}$$

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  We can now easily see that the movie theater charges $2 more for each ticket type at night.
  ::我们现在很容易看出 电影院每张票 晚上每张票多收2美元

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  Example 2
  ::例2

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  Perform the indicated operation.
  

  $$\begin{array}{r}\left[\begin{array}{cc}3& -7\end{array}\right]+\left[\begin{array}{cc}-1& 8\end{array}\right]\end{array}$$

  
  ::[3-7]+[-18]

  
  

  $$\begin{array}{r}\left[\begin{array}{cc}3& -7\end{array}\right]+\left[\begin{array}{cc}-1& 8\end{array}\right]=\left[\begin{array}{cc}3+(-1)& -7+8\end{array}\right]=\left[\begin{array}{cc}2& 1\end{array}\right]\end{array}$$

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  Example 3
  ::例3

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  Perform the indicated operation.
  

  $$\begin{array}{r}\left[\begin{array}{c}1\\ -5\end{array}\right]-\left[\begin{array}{cc}3& -3\\ 4& 1\end{array}\right]\end{array}$$

  
  ::[1-5] -[3-341]

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  These matrices cannot be subtracted because they have different dimensions.
  ::这些矩阵因具有不同维度而无法减去。

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  Example 4
  ::例4

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  Perform the indicated operation.
  

  $$\begin{array}{r}\left[\begin{array}{cc}6& -7\\ -11& 5\end{array}\right]-\left[\begin{array}{cc}-2& 4\\ -3& 9\end{array}\right]\end{array}$$

  
  ::[6-7-115]-[-24-39]

  
  

  $$\begin{array}{r}\left[\begin{array}{cc}6& -7\\ -11& 5\end{array}\right]-\left[\begin{array}{cc}-2& 4\\ -3& 9\end{array}\right]=\left[\begin{array}{cc}6-(-2)& -7-4\\ -11-(-3)& 5-9\end{array}\right]=\left[\begin{array}{cc}8& -11\\ -8& -4\end{array}\right]\end{array}$$

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  Review
  ::回顾

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  Perform the indicated operation (if possible).
  ::执行指示的操作(如果可能)。

  1. .

  $$\begin{array}{r}\left[\begin{array}{cc}2& -1\\ 5& 0\end{array}\right]+\left[\begin{array}{cc}-6& 0\\ 3& -4\end{array}\right]\end{array}$$

  2. .

  $$\begin{array}{r}\left[\begin{array}{cc}3& -2\\ -5& 1\\ 10& 9\end{array}\right]-\left[\begin{array}{cc}-2& 7\\ 10& -8\\ 7& 5\end{array}\right]\end{array}$$

  3. .

  $$\begin{array}{r}\left[\begin{array}{c}4\\ -2\\ 12\\ 7\end{array}\right]+\left[\begin{array}{c}-1\\ 9\\ -2\\ 0\end{array}\right]\end{array}$$

  4. .

  $$\begin{array}{r}\left[\begin{array}{cccc}-1& -4& -1& 12\\ 2& 6& 14& 5\end{array}\right]-\left[\begin{array}{cc}-3& 1\\ 7& -6\end{array}\right]\end{array}$$

  5. .

  $$\begin{array}{r}\left[\begin{array}{cc}4& -1\end{array}\right]+\left[\begin{array}{cc}0& 5\end{array}\right]-\left[\begin{array}{cc}-12& 3\end{array}\right]\end{array}$$

  6. .

  $$\begin{array}{r}\left[\begin{array}{cc}3& 5\end{array}\right]+\left[\begin{array}{cc}-2& -1\end{array}\right]\end{array}$$

  7. .

  $$\begin{array}{r}\left[\begin{array}{c}2\\ 7\end{array}\right]+\left[\begin{array}{cc}-3& 5\end{array}\right]\end{array}$$

  8. .

  $$\begin{array}{r}\left[\begin{array}{ccc}11& 7& -3\\ 9& 15& 8\end{array}\right]+\left[\begin{array}{ccc}20& -4& 7\\ 1& 11& -13\end{array}\right]\end{array}$$

  9. .

  $$\begin{array}{r}\left[\begin{array}{c}25\\ 19\\ -5\end{array}\right]-\left[\begin{array}{c}11\\ 20\\ -3\end{array}\right]\end{array}$$

  10. .

  $$\begin{array}{r}\left[\begin{array}{ccc}2& -5& 3\\ 9& 15& 8\\ -1& -4& 6\end{array}\right]+\left[\begin{array}{ccc}-3& 8& -3\\ 11& -6& -7\\ 0& 8& 5\end{array}\right]\end{array}$$

  11. .

  $$\begin{array}{r}\left[\begin{array}{cc}-3& 2\\ 4& -1\end{array}\right]-\left[\begin{array}{ccc}6& -11& 13\\ 17& 8& 10\end{array}\right]\end{array}$$

  12. .

  $$\begin{array}{r}\left[\begin{array}{cc}-5& 2\\ 9& -3\end{array}\right]+\left[\begin{array}{cc}-3& -5\\ 8& 12\end{array}\right]\end{array}$$

  13. .

  $$\begin{array}{r}\left(\left[\begin{array}{cc}5& -2\\ -3& 1\end{array}\right]+\left[\begin{array}{cc}-8& 5\\ 6& 13\end{array}\right]\right)-\left[\begin{array}{cc}-10& 8\\ 9& 1\end{array}\right]\end{array}$$

  14. .

  $$\begin{array}{r}\left[\begin{array}{cc}-5& 2\\ 11& 3\end{array}\right]-\left(\left[\begin{array}{cc}8& -2\\ 3& 5\end{array}\right]+\left[\begin{array}{cc}-12& 3\\ -6& 15\end{array}\right]\right)\end{array}$$

  15. .

  $$\begin{array}{r}\left(\left[\begin{array}{cc}22& -7\\ 5& 3\\ 11& -8\end{array}\right]-\left[\begin{array}{cc}-8& 9\\ 15& 12\\ 10& -1\end{array}\right]\right)+\left[\begin{array}{cc}5& 11\\ 17& -3\\ -9& 4\end{array}\right]\end{array}$$

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  Review (Answers)
  ::回顾(答复)

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  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 。