4.6 矩阵技术操作

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  采用技术的矩阵操作

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

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  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假日周末上出售的电影院每张门票的数量。

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  Tickets Sold
  ::售票

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  $$\begin{array}{rl}& Sat.Sun.Mon.\\ \begin{array}{c}\text{Kids }\\ \text{Adults }\\ \text{Seniors }\end{array}& \left[\begin{array}{ccc}122& 133& 150\\ 89& 75& 101\\ 57& 38& 49\end{array}\right]\end{array}$$

  
  ::[12.21331508975101573849]

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  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?

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  Matrix Operations
  ::矩阵业务

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  To use the calculator to perform matrix operations, students must first learn how to put matrices into their calculator. If you are using a TI-83 or TI-84, access the Matrix menu by either pressing MATRIX or ( $\begin{array}{r}{2}^{nd}{x}^{-1}\end{array}$ MATRIX). Now you can choose to EDIT matrix $\begin{array}{r}A\end{array}$ . Change the dimensions as needed and enter the data values. You can enter values for a second matrix by selecting matrix $\begin{array}{r}B\end{array}$ in the EDIT menu and repeating the process. Once the matrices have been entered, they can be called up to put in an equation on the calculator by choosing the matrix in the NAME menu. If you do not have a TI-83 or 84, the commands might be different. Check with your teacher.
  ::要使用计算器来进行矩阵操作, 学生必须首先学会如何将矩阵放入计算器。 如果您正在使用 TI- 83 或 TI- 84, 可以按 MATRIX 或 (2nd x- 1 MATRIX) 访问矩阵菜单。 现在您可以选择 EDIT 矩阵 A。 需要时修改尺寸并输入数据值。 您可以在 EDIT 菜单中选择 矩阵 B 并重复此进程, 输入第二个矩阵的值。 一旦输入该矩阵, 就可以在计算器上输入等式, 在 NAME 菜单中选择矩阵 。 如果您没有 TI- 83 或 84, 命令可能会不同 。 请与您的老师检查 。

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  Let's multiply the following matrices together.
  ::让我们一起乘以以下矩阵。

  1.     
     
     $$\begin{array}{r}\frac{2}{3}\left[\begin{array}{cc}5& 6\\ -9& 2\end{array}\right]\end{array}$$

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  This is an example of scalar multiplication . Here we can multiply each of the elements of the matrix by $\begin{array}{r}\frac{2}{3}\end{array}$ by hand or we can use the calculator. To use the calculator, we must first enter the matrix.
  ::这是计算乘法的一个例子。 在这里, 我们可以手工将矩阵的每个元素乘以 23 , 或者使用计算器。 要使用计算器, 我们必须先输入矩阵 。

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  Access the MATRIX menu (press $\begin{array}{r}{2}^{nd}{x}^{-1}\end{array}$ ), select EDIT and matrix $\begin{array}{r}A\end{array}$ . Make the dimensions of the matrix $\begin{array}{r}2\times 2\end{array}$ and then enter the values. Now, exit ( $\begin{array}{r}{2}^{nd}\end{array}$ QUIT) to return to the home screen. We can now type in our expression by typing $\begin{array}{r}\frac{2}{3}[A]\end{array}$ (you can call up matrix $\begin{array}{r}A\end{array}$ in the matrix menu by selecting NAME $\begin{array}{r}[A]\end{array}$ and ENTER) and press ENTER to get the result. The result can then be put into fraction form by pressing MATH and selecting FRAC. The final reduced fraction form is
  ::访问 MATRIX 菜单( press 2nd x-1) , 选择 EDIT 和 矩阵 A。 使矩阵 2x2 的维度再输入值。 现在, 退出 2nd QUIT) 返回主屏幕 。 我们现在可以输入 23 [A] (您可以在矩阵菜单中调用 矩阵 A (通过选择 NAME [A] 和 ENTER) , 按 ENTER 获取结果。 然后, 通过按 MATH 键和选择 FRAC , 将结果以分数形式显示 。 最后减少的分数形式是 :

  $$\begin{array}{r}\left[\begin{array}{cc}\frac{10}{3}& 4\\ -6& \frac{4}{3}\end{array}\right].\end{array}$$

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

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  This multiplication problem would be very tedious to perform by hand. We can use the calculator to make the process much more efficient and avoid making careless arithmetic errors as well.
  ::这个乘法问题如果用手来操作,将会非常烦琐。 我们可以使用计算器来提高计算过程的效率,避免粗略的算术错误。

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  Go to the MATRIX menu on your calculator and EDIT matrix $\begin{array}{r}A\end{array}$ . Note that the dimensions should be $\begin{array}{r}3\times 4\end{array}$ . Now, enter the elements of the matrix. When you are finished, exit ( $\begin{array}{r}{2}^{nd}\end{array}$ QUIT) to return to the home screen. Return to the MATRIX menu and EDIT matrix $\begin{array}{r}B\end{array}$ . Note that the dimensions of matrix $\begin{array}{r}B\end{array}$ are $\begin{array}{r}4\times 3\end{array}$ . Enter the data elements and return to the home screen. Now we can multiply the matrices together by calling them up from the MATRIX menu. Return to the MATRIX menu and select NAME $\begin{array}{r}[A]\end{array}$ . You will be automatically returned to the home where $\begin{array}{r}[A]\end{array}$ should be showing. Now, return to the MATRIX menu and select NAME $\begin{array}{r}[B]\end{array}$ . Your home screen should now have $\begin{array}{r}[A][B]\end{array}$ . It is not necessary to put a multiplication symbol between the two matrices since the brackets act like parenthesis and already indicate multiplication. Press ENTER and the result will appear:
  ::转到您的计算器和 EDIT 矩阵 A 上的 MATRIX 菜单。 请注意, 尺寸应该是 3x4 。 现在, 输入矩阵的元素。 完成后, 退出 2( QUIT) 返回主屏幕 。 返回 MATRIX 菜单和 EDIT 矩阵 B 。 注意 矩阵 B 的尺寸是 4x3. 输入数据元素并返回主屏幕 。 现在, 我们可以从 MATRIX 菜单中将它们调用到 MATRIX 菜单中, 并选择 NAME 。 现在, 返回 MATRIX 菜单并选择 NAME 。 现在, 您的家屏幕应该有 [A [B] 。 不必在两个矩阵之间设置乘法符号, 因为括号像 parheis 一样, 并且已经显示乘法 。 按 ENTER 键和结果将显示 :

  $$\begin{array}{r}\left[\begin{array}{ccc}51& 3& -2\\ 25& -11& -31\\ 48& -6& -30\end{array}\right].\end{array}$$

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  Now, let's evaluate the following expression:

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

  
  ::现在,让我们来评价以下表达式:12[8-351][-4167]+[-2-190]

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  This time we will need to enter three matrices into the calculator. Let’s let
  ::这次我们需要将三个矩阵输入计算器。

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

  Once you have entered these matrices, go to the home screen and type in the expression: $\begin{array}{r}\frac{1}{2}[A][B]+[C]\end{array}$ by calling up the matrices from the MATRIX menu as needed. Now press ENTER and MATH FRAC to get the final result:

  $$\begin{array}{r}\left[\begin{array}{cc}-27& -\frac{15}{2}\\ 2& 6\end{array}\right].\end{array}$$

  
  ::[A]=[8-351],[B]=[-4167],[C]=[-2-190]。您进入这些矩阵后,进入主屏幕并输入表达式:12[A][B]+[C],根据需要从MATRIX菜单中调用矩阵。现在按 ENTER 和MATH FRAC 键以获得最后结果:[-27-15226]。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked to find  the total amount of money the movie theater t ook in for the three day weekend. 
  ::早些时候,你被要求去寻找 电影院在三天周末 收下的钱总额。

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  This is a problem where the matrix is multiplied by a scalar.
  ::当矩阵乘以一个标标量时,这是一个问题。

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  Tickets Sold
  ::售票

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  $$\begin{array}{rl}& Sat.Sun.Mon.\\ \begin{array}{c}\text{Kids }\\ \text{Adults }\\ \text{Seniors }\end{array}5\cdot & \left[\begin{array}{ccc}122& 133& 150\\ 89& 75& 101\\ 57& 38& 49\end{array}\right]\end{array}$$

  
  ::[1221331508975101573849]

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  Use your calculator to assist you. The resulting matrix is:
  ::使用计算器来帮助您。 由此得出的矩阵是 :

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  Tickets Sales ($)

  $$\begin{array}{rl}& Sat.Sun.Mon.\\ \begin{array}{c}\text{Kids }\\ \text{Adults }\\ \text{Seniors }\end{array}\left[\begin{array}{ccc}610& 665& 750\\ 445& 375& 505\\ 285& 190& 245\end{array}\right]\end{array}$$

  
  ::儿童/成年人/老年人 = [6106657504557575755285190245]

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  From this matrix, we can see that the movie theater made $610 from kids' ticket sales on Saturday, $665 on Sunday, and $750, so the total amount made from kid's ticket sales was $\begin{array}{r}\mathrm{\$}610+\mathrm{\$}665+\mathrm{\$}750=\mathrm{\$}2025\end{array}$ .
  ::从这个矩阵中,我们可以看到,电影院从星期六的儿童票销售中赚到了610美元,星期天665美元,750美元,因此,儿童票销售总额是610美元+665美元+750美元=2025美元。

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  Use your calculator to evaluate the following matrix expressions.
  ::使用您的计算器来评价以下矩阵表达式 。

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  Example 2
  

  $$\begin{array}{r}\frac{1}{2}\left[\begin{array}{cc}4& -3\end{array}\right]\left[\begin{array}{cc}8& 4\\ -6& -2\end{array}\right]\end{array}$$

  
  ::例2 12[4-3][84-6-2]

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  Let $\begin{array}{r}[A]=[4-3]\end{array}$ and

  $$\begin{array}{r}[B]=\left[\begin{array}{cc}8& 4\\ -6& -2\end{array}\right].\end{array}$$

  Now we can go to the home screen and type in the expression: $\begin{array}{r}\frac{1}{2}[A][B]\end{array}$ and press ENTER to get: $\begin{array}{r}[2511]\end{array}$ .
  ::让我们[A]=[4-3]和[B]=[84-6-2]。现在我们可以去主屏幕上输入:12[A][B]和按ENTER,以获得:[2511]。

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  Example 3
  ::例3

  
  

  $$\begin{array}{r}\left[\begin{array}{cc}1& -2\\ 3& -4\end{array}\right]\left(\left[\begin{array}{c}2\\ -5\end{array}\right]+\left[\begin{array}{c}-4\\ 8\end{array}\right]\right)\end{array}$$

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   Let

  $$\begin{array}{r}[A]=\left[\begin{array}{cc}1& -2\\ 3& -4\end{array}\right],[B]=\left[\begin{array}{c}2\\ -5\end{array}\right]\end{array}$$

  and

  $$\begin{array}{r}[C]=\left[\begin{array}{c}-4\\ 8\end{array}\right].\end{array}$$

  Return to the home screen and type in the expression: $\begin{array}{r}[A]([B]+[C])\end{array}$ and press ENTER to get: ERR: INVALID DIM. We know that it is, in fact, possible to add matrix $\begin{array}{r}B\end{array}$ and matrix $\begin{array}{r}C\end{array}$ and multiply the result by matrix $\begin{array}{r}A\end{array}$ on the left.
  ::Let[A]=[1-23-4]、[B]=[2-5]和[C]=[-48]。返回主屏幕并输入表达式:[A]([B]+[C])和按ENTER键获得:ERR:INVALID DIM。我们知道,事实上可以添加矩阵B和矩阵C,并将结果乘以左边的矩阵A。

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  $\begin{array}{r}{}^{\ast }\end{array}$ We need to make sure that we multiply $\begin{array}{r}[A]\end{array}$ on the left of $\begin{array}{r}[B]+[C]\end{array}$ because multiplication of matrices is not commutative (and because the dimensions would not work for multiplication the other way around because we would have a $\begin{array}{r}2\times 1\end{array}$ multiplied by a $\begin{array}{r}2\times 2\end{array}$ .)
  ::* 我们需要确保在[B]+[C]的左侧乘以[A],因为矩阵的乘法不是通量的(并且因为尺寸不会对反过来的乘法起作用,因为我们的乘法是2x1乘以2x2)

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  Let’s try breaking this one down into these two steps on the calculator:
  ::让我们尝试将这一步分成两个计算器的两步:

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  Step 1: Add $\begin{array}{r}[B]+[C]\end{array}$
  ::第1步:添加[B]+[C]

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  Step 2: Now, call up matrix $\begin{array}{r}A\end{array}$ and multiply by the answer from step 1: $\begin{array}{r}[A]\end{array}$ Ans (to get Ans, press $\begin{array}{r}{2}^{nd}\end{array}$ and (-) button.) for the result:

  $$\begin{array}{r}\left[\begin{array}{c}-8\\ -18\end{array}\right].\end{array}$$

  
  ::第2步:现在,调用矩阵A,然后乘以第1步的答案:[A] 答复(获得答复,按2和(-)按钮),结果为:[-8-18]。

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  Reason for the initial error: The calculator must be trying to multiply the sum of matrices $\begin{array}{r}B\end{array}$ and $\begin{array}{r}C\end{array}$ by matrix $\begin{array}{r}A\end{array}$ on the right and this product is not possible due to the mismatched dimensions.
  ::最初误差的原因:计算器必须设法将基数B和C的总和乘以右边的基数A,而由于不匹配的维度,这一产品是不可能的。

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  Example 4
  

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

  
  ::例4 [25-38-42-706][7-31]+5[4-82]

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  Let

  $$\begin{array}{r}[A]=\left[\begin{array}{ccc}2& 5& -3\\ 8& -4& 2\\ -7& 0& 6\end{array}\right],[B]=\left[\begin{array}{c}7\\ -3\\ 1\end{array}\right]\end{array}$$

  and

  $$\begin{array}{r}[C]=\left[\begin{array}{c}4\\ -8\\ 2\end{array}\right].\end{array}$$

  Now we can return to the home screen and type in the expression: $\begin{array}{r}[A][B]+5[C]\end{array}$ . Now press ENTER to get:

  $$\begin{array}{r}\left[\begin{array}{c}16\\ 30\\ -33\end{array}\right].\end{array}$$

  
  ::[A]=[25-38-42-706],[B]=[7-31]和[C]=[4-82]。现在我们可以回到主屏幕上,并输入[A][B]+5[C]。现在按ENTER获取:[1630-33]。

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  Review
  ::回顾

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  Use a calculator to evaluate the following expressions, if possible.
  ::在可能的情况下,使用计算器来评价以下表达式。

  1. .

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

  2. .

  $$\begin{array}{r}\left[\begin{array}{ccc}4& -1& 2\\ -7& 9& 0\\ 5& -6& 1\end{array}\right]\cdot \left[\begin{array}{cccc}-2& 5& 1& -3\\ -3& 9& 10& -1\\ 0& 8& 11& 1\end{array}\right]\end{array}$$

  3. .

  $$\begin{array}{r}\frac{1}{3}\left[\begin{array}{ccc}3& -1& 8\\ 2& 0& 5\end{array}\right]\cdot \left[\begin{array}{c}2\\ -1\\ 7\end{array}\right]\end{array}$$

  4. .

  $$\begin{array}{r}-2\left[\begin{array}{cc}0& 1\\ 2& 0\end{array}\right]\cdot \left[\begin{array}{cc}2& -5\end{array}\right]\end{array}$$

  5. .

  $$\begin{array}{r}\left(\left[\begin{array}{ccc}2& -3& 7\end{array}\right]+\left[\begin{array}{ccc}-3& 5& 1\end{array}\right]\right)\cdot \left[\begin{array}{c}-4\\ 8\\ 11\end{array}\right]\end{array}$$

  6. .

  $$\begin{array}{r}\left[\begin{array}{c}4\\ -3\end{array}\right]-\left[\begin{array}{cc}5& -2\\ 7& 11\end{array}\right]\cdot \left[\begin{array}{c}-6\\ 8\end{array}\right]\end{array}$$

  7. .

  $$\begin{array}{r}\left[\begin{array}{cccc}2& -6& 5& 7\\ 8& 11& -1& -3\\ -4& 9& 10& 12\end{array}\right]\cdot \left[\begin{array}{ccc}-11& -6& 1\\ 7& 10& 3\\ 8& 4& 5\\ 5& 2& 6\end{array}\right]\end{array}$$

  8. .

  $$\begin{array}{r}\left[\begin{array}{ccc}4& 1& -2\\ -2& 1& 6\\ -12& 4& 10\end{array}\right]\cdot 4\left[\begin{array}{c}3\\ 8\\ -9\end{array}\right]\end{array}$$

  9. .

  $$\begin{array}{r}\frac{1}{5}\left[\begin{array}{c}1\\ 3\\ 7\end{array}\right]\cdot \left[\begin{array}{ccc}-2& 8& 4\end{array}\right]\end{array}$$

  10. .

  $$\begin{array}{r}\left[\begin{array}{cccc}-2& 1& 6& 3\\ 3& 0& 12& 4\\ 7& -5& -8& -12\\ -4& 11& 1& 0\end{array}\right]\cdot \left[\begin{array}{cccc}5& -3& 1& 6\\ -4& 2& 13& 11\\ 6& 7& 5& 3\\ 10& -8& -6& 4\end{array}\right]\end{array}$$

  11. .

  $$\begin{array}{r}\left[\begin{array}{cc}2& -8\\ 12& 5\\ -3& 11\end{array}\right]\cdot \left[\begin{array}{ccc}4& -6& 10\\ -2& 8& 7\end{array}\right]\end{array}$$

  12. .

  $$\begin{array}{r}\frac{1}{3}\left[\begin{array}{c}8\\ -7\end{array}\right]+\frac{1}{2}\left[\begin{array}{ccc}-6& 3& -5\\ 2& 11& 8\end{array}\right]\cdot \left[\begin{array}{c}2\\ -1\\ 7\end{array}\right]\end{array}$$

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  For problems 13 and 14, the calculator will give you an error. Can you explain why? Is it an error you can work around or is the problem just not possible? If you can work around the issue, complete the problem.
  ::对于问题13和14, 计算器会给您一个错误。 您能解释原因吗 ? 您能解释原因吗 ? 是您可以绕圈子工作的错误还是问题不可能解决? 如果您能绕圈子工作, 请填满问题 。

  13. .

  $$\begin{array}{r}\left[\begin{array}{cc}2& -3\end{array}\right]\left(\left[\begin{array}{cc}-1& 11\\ 3& -7\end{array}\right]+\left[\begin{array}{cc}4& -2\\ 0& 9\end{array}\right]\right)\end{array}$$

  14. .

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

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  15. Real world application: A movie theater tracks the sales 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 second matrix. How much revenue did the theater take in each night for popcorn? How much did the theater take in for popcorn in total?
      $$\begin{array}{rl}& SML\text{Price}\\ \begin{array}{c}\text{Friday }\\ \text{Saturday }\\ \text{Sunday }\end{array}& \left[\begin{array}{ccc}36& 85& 40\\ 41& 112& 51\\ 28& 72& 35\end{array}\right]\begin{array}{c}S\\ M\\ L\end{array}\left[\begin{array}{c}5.50\\ 6.25\\ 7.25\end{array}\right]\end{array}$$
      
      ::真实世界应用程序: 一个电影院以三种不同大小追踪爆米花的销售情况。 在周末(星期五、星期六和星期天晚上)收集的数据在下面的矩阵中显示。 每种大小的价格在第二个矩阵中显示。 每晚剧院在爆米花上收多少收入? 剧院在爆米花上总共收了多少? SM L PriceFriday\星期六\星期天\ [36854041111251287235] SML [5.506.257.25] SML [5.50.6257.25]

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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 。