4.5 矩阵乘法的局限性

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  矩阵乘法的局限性

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  Mr. Hwan writes the following matrix on the board and asks his students to write down a matrix that they could multiply this matrix by.
  ::Hwan先生在董事会上写下以下表格,并请他的学生写下一个表格,以便他们可以乘以这一表格。

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

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  Wanda writes the matrix

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

  
  ::Wanda写了汇总表[-12530-2]

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  Xavier writes the matrix

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

  
  ::Xavier 写下矩阵 [2-371]

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  Zach writes the matrix

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

  
  ::Zach写了矩阵[-41085-3]。

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  "One of you has made a mistake," Mr. Hwan says. Who made the mistake?
  ::万先生说"你们其中一人犯了错" 谁犯了错?

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  Limitations to Matrix Multiplication
  ::矩阵乘法的局限性

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  Matrix multiplication has some limitations, which you will discover in  this section.
  ::矩阵乘法有一些局限性,你会在本节中发现。

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  Note the dimensions of the following matrices. Attempt to multiply the matrices together. What do you notice?
  ::注意以下矩阵的维度。 试图将矩阵乘在一起。 您注意到什么 ?

  1.

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

  2.

  $$\begin{array}{r}\left[\begin{array}{c}-2\\ 3\\ 5\end{array}\right]\cdot \left[\begin{array}{c}4\\ -1\\ 0\end{array}\right]\end{array}$$

  3.

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

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  1. These matrices are $\begin{array}{r}2\times 2\end{array}$ and $\begin{array}{r}1\times 2\end{array}$ . When we try to multiply them together, there is no element to multiply the values in the second column by as shown below:
  ::1. 这些矩阵是 2x2 和 1x2。 当我们试图将它们相乘时,没有元素可以乘以第二列中的数值,具体如下:

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

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  2. These matrices are both $\begin{array}{r}3\times 1\end{array}$ . Look what happens when we attempt to multiply:
  ::2. 这些矩阵均为3x1。

  $$\begin{array}{r}\left[\begin{array}{c}-2\\ 3\\ 5\end{array}\right]\cdot \left[\begin{array}{c}4\\ -1\\ 0\end{array}\right]=\left[\begin{array}{cc}-2(4)+?(-1)+?(0)& \\ \\ \end{array}\right]\end{array}$$

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  We can see right away that we have a problem.
  ::我们马上可以看到,我们有一个问题。

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  3. In this case we have a $\begin{array}{r}2\times 1\end{array}$ and a $\begin{array}{r}2\times 2\end{array}$ matrix. Again we have trouble multiplying:
  ::3. 在这种情况下,我们有一个2x1和2x2矩阵。

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

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  Therefore , i n order to multiply two matrices, $\begin{array}{r}A\end{array}$ and $\begin{array}{r}B\end{array}$ , the number of columns in matrix $\begin{array}{r}A\end{array}$ must be equal to the number of rows in column B.
  ::因此,为了将两个矩阵(A和B)乘以,矩阵A的列数必须与列B的行数相等。

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  What do you notice about the dimensions of the two matrices when multiplication is possible? Can you make a conjecture about the requirements of the dimensions of the matrices?
  ::当可以乘法时,你注意到两个矩阵的尺寸是什么?你能猜测矩阵的尺寸要求吗?

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  Look at this example:

  $$\begin{array}{r}\left[\begin{array}{cc}-2& 3\\ 1& 5\end{array}\right]\cdot \left[\begin{array}{c}4\\ -1\end{array}\right].\end{array}$$

  In this case we have $\begin{array}{r}2\times 2\cdot 2\times 1\end{array}$ . The two two’s in the middle are the same (i.e., the number of columns in the first matrix matches the number of rows in the second matrix). Now what is left is the 2 on the left and the 1 on the right. The resulting matrix will have these dimensions: $\begin{array}{r}2\times 1\end{array}$ .
  ::看看这个例子:[- 2315] 和[ 4- 1]。 在这种情况下,我们有两个在中间的2x2x2xx1。 中间的两个是相同的( 即第一个矩阵中的列数与第二个矩阵中的行数相符 ) 。 现在剩下的是左边的2个和右边的1个。 由此产生的矩阵将具有这些维度: 2x1 。

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  Dimensions of the Product: The product of two matrices, $\begin{array}{r}A\end{array}$ and $\begin{array}{r}B\end{array}$ , will have the following dimensions – the rows in matrix $\begin{array}{r}A\end{array}$ by the columns in matrix $\begin{array}{r}B\end{array}$ .
  ::产品的尺寸:两个矩阵(A和B)的产物将具有以下尺寸—— 矩阵A中的行和矩阵B中的列。

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  Let's use the following matrices  to answer the problems below.
  ::让我们使用以下矩阵来回答下面的问题。

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  $$\begin{array}{r}A=\left[\begin{array}{cc}-2& 5\\ 1& 3\end{array}\right],B=\left[\begin{array}{cc}1& 3\\ 0& -5\end{array}\right]\end{array}$$

  
  ::A=[-2513],B=[130-5]

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  1. Find $\begin{array}{r}AB\end{array}$ :

  $$\begin{array}{r}\left[\begin{array}{cc}-2& 5\\ 1& 3\end{array}\right]\cdot \left[\begin{array}{cc}1& 3\\ 0& -5\end{array}\right]=\left[\begin{array}{cc}-2+0& -6-25\\ 1+0& 3-15\end{array}\right]=\left[\begin{array}{cc}-2& -31\\ 1& -12\end{array}\right]\end{array}$$

  
  ::1. 发现AB:[-2513]-[130-5]=[-2+0-6-6-251+03-15]=[-2-311-12]

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  2. Find $\begin{array}{r}BA\end{array}$ :

  $$\begin{array}{r}\left[\begin{array}{cc}1& 3\\ 0& -5\end{array}\right]\cdot \left[\begin{array}{cc}-2& 5\\ 1& 3\end{array}\right]=\left[\begin{array}{cc}-2+3& 5+9\\ 0-5& 0-15\end{array}\right]=\left[\begin{array}{cc}1& 14\\ -5& -15\end{array}\right]\end{array}$$

  
  ::2. 查找BA:[130-5]-[-2513]=[-2+35+90-50-15]=[114-5-15]

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  From this exercise we can see that $\begin{array}{r}AB\ne BA\end{array}$ . We only need one example to show that the commutative property does not hold for matrix multiplication.
  ::从这一练习中,我们可以看到ABBA。我们只需要一个例子来证明通货财产不用于矩阵乘法。

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  Matrix multiplication is NOT commutative.
  ::矩阵乘法非通量。

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  Let's answer the following questions about matrix multiplication.
  ::让我们回答以下关于矩阵乘法的问题。

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  1. Which of the following matrix multiplication problems are possible? What are the dimensions of the product?
     ::以下矩阵的哪些乘法问题是可能的?产品的尺寸是多少?

  
  

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

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  Yes, the matrices can be multiplied. The dimensions of the matrices are $\begin{array}{r}3\times 1\text{and}1\times 3\end{array}$ , so the result is going to be a $\begin{array}{r}3\times 3\end{array}$ matrix.
  ::是的, 矩阵可以乘以。 矩阵的尺寸是 3x1 和 1x3, 因此结果将是 3x3 矩阵 。

  
  

  $$\begin{array}{r}\left[\begin{array}{cc}4& -1\\ 2& 3\end{array}\right]\cdot \left[\begin{array}{c}-8\\ 0\end{array}\right]\end{array}$$

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  Yes, the matrices can be multiplied. The dimensions of the matrices are $\begin{array}{r}2\times 2\text{and}2\times 1\end{array}$ , so the result is going to be a $\begin{array}{r}2\times 1\end{array}$ matrix.
  ::是的, 矩阵可以乘以。 矩阵的尺寸是 2x2 和 2x1, 因此结果将是 2x1 矩阵 。

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  2. Which of the following matrix multiplication problems are possible? What are the dimensions of the product?
     ::以下矩阵的哪些乘法问题是可能的?产品的尺寸是多少?

  
  

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

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  Yes, the matrices can be multiplied. The dimensions of the matrices are $\begin{array}{r}1\times 2\text{and}2\times 1\end{array}$ , so the result is going to be a $\begin{array}{r}1\times 1\end{array}$ matrix.
  ::是的, 矩阵可以乘以。 矩阵的尺寸是 1x2 和 2x1, 因此结果将是 1x1 矩阵 。

  
  

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

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  No, the matrices cannot be multiplied. The dimensions of the matrices are $\begin{array}{r}1\times 2\text{and}1\times 2\end{array}$ , the number of columns in the first matrix does not match the number of rows in the second matrix.
  ::否, 矩阵不能乘以。 矩阵的尺寸为 1x2 和 1x2, 第一个矩阵的列数与第二个矩阵的行数不符。

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  3. Given the matrices:
     $$\begin{array}{r}A=\left[\begin{array}{ccc}2& -1& 8\\ 3& 0& -5\end{array}\right]\end{array}$$
     and
     $$\begin{array}{r}B=\left[\begin{array}{cc}-1& 0\\ 2& 4\\ 5& 6\end{array}\right],\end{array}$$
     find $\begin{array}{r}AB\end{array}$ and $\begin{array}{r}BA\end{array}$ . What do you notice?
     ::鉴于矩阵:A=[2-1830-5]和B=[-102456],请找到AB和BA。

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  $$\begin{array}{rl}AB=\left[\begin{array}{ccc}2& -1& 8\\ 3& 0& -5\end{array}\right]\cdot \left[\begin{array}{cc}-1& 0\\ 2& 4\\ 5& 6\end{array}\right]& =\left[\begin{array}{cc}2(-1)+(-1)(2)+8(5)& 2(0)+(-1)(4)+8(6)\\ 3(-1)+(0)(2)+(-5)(5)& 3(0)+(0)(4)+(-5)(6)\end{array}\right]\\ & =\left[\begin{array}{cc}-2-2+40& 0-4+48\\ -3+0-25& 0+0-30\end{array}\right]=\left[\begin{array}{cc}36& 44\\ -28& -30\end{array}\right]\end{array}$$

  
  ::AB=[2-1830-55][-102456]=[(-1)+(-1)(2)+8(5)2(0)+(-1)(2)+(-1)(4)(4)+8(6)3(-1)+(1)+(0)(2)+(-5)(5)3(0)+(0)(4)+(-5)(6)]=[-2-2-2+4004+4-48-3+0-250+0-0-30]=[3644-28-30]

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  $$\begin{array}{rl}BA=\left[\begin{array}{cc}-1& 0\\ 2& 4\\ 5& 6\end{array}\right]\cdot \left[\begin{array}{ccc}2& -1& 8\\ 3& 0& -5\end{array}\right]& =\left[\begin{array}{ccc}-1(2)+0(3)& -1(-1)+0(0)& -1(8)+0(-5)\\ 2(2)+4(3)& 2(-1)+4(0)& 2(8)+4(-5)\\ 5(2)+6(3)& 5(-1)+6(0)& 5(8)+6(-5)\end{array}\right]\\ & =\left[\begin{array}{ccc}-2+0& 1+0& -8-0\\ 4+12& -2+0& 16-20\\ 10+18& -5+0& 40-30\end{array}\right]=\left[\begin{array}{ccc}-2& 1& -8\\ 16& -2& -4\\ 28& -5& 10\end{array}\right]\end{array}$$

  
  ::BA=[-102456]-[2 - 1830-5]=[-1(2)+0(3)-1(-1)+0(3)-1(0)-1(1)-0(0)-1(8)+0(5)(5)(5)(2)(2)+4(2)(1)+4(0)(5)(5)(5)(2)+6(3)5(1)+6(0)5(8)+6(5)]=[-2+01+0-004+0-804+12-2+016-2010+18-18+5+040-30]=[-21-816-2-428-510]

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  The dimensions of the products are different based on the order in which we multiply them. Clearly, the results are different and the commutative property does not hold for matrix multiplication.
  ::根据我们乘以它们的顺序,产品的尺寸是不同的。 显然,结果是不同的,通货性能不适于矩阵乘法。

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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 student that made the mistake.
  ::早些时候,有人要求你找到犯错误的学生。

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  Mr. Hwan's matrix is a 3 x 2 matrix. To multiply it by another matrix, that other matrix must be a 2 x ? matrix, where ? can be any number.
  ::Hwan先生的矩阵是一个 3 x 2 矩阵,要将其乘以另一个矩阵,其他矩阵必须是 2 x? 矩阵,在哪里可以有数字。

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  Wanda writes a 3 x 2 matrix, so she is wrong.
  ::Wanda写了一个3x2矩阵,所以她错了。

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  Xavier writes a 2 x 2 matrix, so he is correct.
  ::Xavier写了一个 2 x 2 矩阵, 所以他是正确的 。

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  Zach writes a 2 x 3 matrix, so he is correct.
  ::Zach写了一个 2 x 3 矩阵, 所以他是正确的。

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  Given:

  $$\begin{array}{r}A=\left[\begin{array}{cc}1& -3\\ 5& -14\end{array}\right],B=\left[\begin{array}{c}-2\\ 1\end{array}\right],C=\left[\begin{array}{cc}-14& 3\\ -5& 1\end{array}\right],\end{array}$$

  find the following products, if possible.
  ::A=[1-35-14]、B=[-21]、C=[-143-51],如有可能,可找到下列产品。

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  Example 2
  ::例2

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  $\begin{array}{r}AB\end{array}$
  

  $$\begin{array}{r}AB=\left[\begin{array}{cc}1& -3\\ 5& -14\end{array}\right]\cdot \left[\begin{array}{c}-2\\ 1\end{array}\right]=\left[\begin{array}{c}1(-2)+-3(1)\\ 5(-2)+-14(1)\end{array}\right]=\left[\begin{array}{c}-2-3\\ -10-14\end{array}\right]=\left[\begin{array}{c}-5\\ -24\end{array}\right]\end{array}$$

  
  ::AB AB = [1-35-14] = [1-2] 3(1)5(-2) 14(1)] = [-2-3-10-14] = [-5-24]

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  Example 3
  ::例3

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  $\begin{array}{r}BA\end{array}$
  

  $$\begin{array}{r}BA=\left[\begin{array}{c}-2\\ 1\end{array}\right]\cdot \left[\begin{array}{cc}1& -3\\ 5& -14\end{array}\right]\end{array}$$

  is not possible.
  ::BA=[-21]+[1-35-14]不可能。

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  Example 4
  ::例4

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  $\begin{array}{r}AC\end{array}$
  ::AC( AC) AC

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  $$\begin{array}{r}AC=\left[\begin{array}{cc}1& -3\\ 5& -14\end{array}\right]\cdot \left[\begin{array}{cc}-14& 3\\ -5& 1\end{array}\right]=\left[\begin{array}{cc}1(-14)+(-3)(-5)& 1(3)+(-3)(1)\\ 5(-14)+(-14)(-5)& 5(3)+(-14)(1)\end{array}\right]=\left[\begin{array}{cc}-14+15& 3-3\\ -70+70& 15-14\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\end{array}$$

  
  ::AC=[1-35-14]][-143-51]=[1(-14)+(-3)+(-3)(-5)1(3)+(-3)(1)5(-14)+(-14)+(-14)(-5)+(-5)+(-14)(1)]=[-14+1533-70-70+7015-14]=[1001]

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  Example 5
  ::例5

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  $\begin{array}{r}CA\end{array}$
  ::CAC 会计师协会

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  $$\begin{array}{r}CA=\left[\begin{array}{cc}-14& 3\\ -5& 1\end{array}\right]\cdot \left[\begin{array}{cc}1& -3\\ 5& -14\end{array}\right]=\left[\begin{array}{cc}-14(1)+3(5)& -14(-3)+3(-14)\\ -5(1)+1(5)& -5(-3)+1(-14)\end{array}\right]=\left[\begin{array}{cc}-14+15& 52-52\\ -5+5& 15-14\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\end{array}$$

  
  ::CA=[-143-51]/[1-35-14]=[-14(1)+3(5)-14(-3-3)、+3(-14)-5(1)+1(5)-5(-3)+1(-14)]=[-14+1552-55+515-14]=[1001]

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  $\begin{array}{r}{}^{\ast \ast }\end{array}$ Note: When the product is

  $$\begin{array}{r}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\end{array}$$

  the identity matrix for a $\begin{array}{r}2\times 2\end{array}$ matrix, the product can be found by multiplying the matrices in either order as illustrated here in questions 3 and 4. If the product of two matrices is the inverse matrix, then the matrices are multiplicative inverses of one another. This idea is further explored in other matrix concepts.
  ::注:当产品为[1001],即2×2矩阵的特性矩阵时,可通过将矩阵按问题3和4所示的任一顺序乘以来找到产品,如果两个矩阵的产物是逆矩阵,那么这些矩阵是相互重复的反变量,在其他矩阵概念中进一步探讨这一想法。

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  Review
  ::回顾

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  Use the following matrices to answer questions 1-10.
  ::使用以下矩阵回答问题1-10。

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  $$\begin{array}{r}A=\left[\begin{array}{ccc}2& -1& 4\end{array}\right]B=\left[\begin{array}{c}-3\\ 5\\ 0\end{array}\right]C=\left[\begin{array}{cc}4& -1\\ 2& 5\end{array}\right]D=\left[\begin{array}{cc}3& -2\end{array}\right]\end{array}$$

  
  ::A=[2-14]B=[-350]C=[4-125]D=[3-2]

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  $$\begin{array}{r}E=\left[\begin{array}{ccc}8& -1& 0\\ 2& 5& 1\\ 3& 4& -3\end{array}\right]F=\left[\begin{array}{ccc}5& 0& -11\\ -1& 8& 9\end{array}\right]\end{array}$$

  
  ::E=[8-1025134-3]F=[50-11-189]

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  For each product, determine the dimensions of the resulting matrix or write “cannot be determined”. Do not multiply the matrices together.
  ::对每种产品,确定所产生的矩阵的尺寸,或写“无法确定”,不要将矩阵相乘。

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  1. $\begin{array}{r}AB\end{array}$
     ::AB
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  2. $\begin{array}{r}BA\end{array}$
     ::学 学 学 学 学 学 学 学 学 学 学 学 学 学 学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学学
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  3. $\begin{array}{r}CD\end{array}$
     ::CD CD CD
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  4. $\begin{array}{r}DC\end{array}$
     ::DC DC
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  5. $\begin{array}{r}EA\end{array}$
     ::职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职 职
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  6. $\begin{array}{r}EB\end{array}$
     ::出 乙
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  7. $\begin{array}{r}CF\end{array}$
     ::CFF 瑞士法郎
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  8. $\begin{array}{r}FC\end{array}$
     ::花花公会
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  9. $\begin{array}{r}DF\end{array}$
     ::DFF 开发基金
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  10. $\begin{array}{r}BD\end{array}$
      ::业 业 业 业 业 业 业 业

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  Perform the multiplication if possible. If not possible, write “cannot be determined”.
  ::如果可能的话, 执行乘法。 如果不可能, 写“ 无法确定 ” 。

  11. .

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

  12. .

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

  13. .

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

  14. .

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

  15. .

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

  16. .

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

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  17. Challenge Problem
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      1. Find
         $$\begin{array}{r}\left[\begin{array}{cc}5& 3\\ 8& 5\end{array}\right]\cdot \left[\begin{array}{cc}5& -3\\ -8& 5\end{array}\right]\end{array}$$
         
         ::[5-3-3-85]
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      2. What is the name of the product matrix?
         ::产品矩阵的名称是什么?
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      3. Can you find
         $$\begin{array}{r}\left[\begin{array}{cc}5& -3\\ -8& 5\end{array}\right]\cdot \left[\begin{array}{cc}5& 3\\ 8& 5\end{array}\right]\end{array}$$
         without actually multiplying?
         ::[5-3-85] [5385] 在不实际乘法的情况下能找到[5-3-85]][5385]吗?
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      4. What can you conclude about the two matrices in part a?
         ::关于A部分的两个矩阵,你能得出什么结论?
      
      ::[5-385] [5-3-3-85] 产品矩阵的名称是什么?您能找到[5-3-3-85] {[5385] 而不实际乘法吗?

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  Review (Answers)
  ::回顾(答复)

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