8.4 矩阵业务

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  矩阵业务

  Algebra refers to your ability to manipulate variables and unknowns based on rules and properties. Matrix algebra is extremely similar to the algebra you already know for numbers with a few important differences.  What are these differences?

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  Algebra with Matrices
  ::数学代数

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  Addition and Subtraction
  ::加和减

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  Two matrices of the same order can be added by summing the entries in the corresponding positions.
  ::可以通过在相应的职位上对条目进行汇总,添加相同顺序的两个矩阵。

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

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  Two matrices of the same order can be subtracted by subtracting the entries in the corresponding positions. 
  ::可以通过减去相应职位中的条目来减去同一顺序的两个矩阵。

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

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  Multiplication
  ::乘法乘法

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  You can find the product of matrix  $\begin{array}{r}A\end{array}$ and matrix  $\begin{array}{r}B\end{array}$ if the number of columns in matrix  $\begin{array}{r}A\end{array}$ matches the number of rows in matrix $\begin{array}{r}B\end{array}$ .  Another way to remember this is when you write the orders of matrix  $\begin{array}{r}A\end{array}$ and matrix  $\begin{array}{r}B\end{array}$ next to each other they must be connected by the same number.  The resulting matrix has the number of rows from the first matrix and the number of columns from the second matrix. 
  ::如果矩阵A和矩阵B的列数与矩阵B的行数相符,您可以找到矩阵A和矩阵B的产物。另一个记住的方法是,当您将矩阵A和矩阵B的顺序相邻写成表格A和矩阵B的顺序时,它们必须用相同的编号连接。由此产生的矩阵含有第一个矩阵的行数和第二个矩阵的列数。

  $\begin{array}{r}(2\times 3)\cdot (3\times 5)=(2\times 5)\end{array}$

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  To compute the first entry of the resulting $\begin{array}{r}2\times 5\end{array}$  matrix you should match the first row from the first matrix and the first column of the second matrix.  The arithmetic operation to combine these numbers is identical to taking the dot product between two vectors.
  ::要计算生成的 2x5 矩阵的第一个条目, 您应该匹配第一个矩阵的第一行和第二个矩阵的第一列。 合并这些数值的算术操作与在两个矢量之间取出点产品相同 。

  

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  • The entry in the first row first column of the new matrix is computed as $\begin{array}{r}1\cdot 0+4\cdot 2+3\cdot 1=11\end{array}$ .
    ::新矩阵第一行第一列条目的计算值为 10+42+31=11。
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  • The entry in the second row first column of the new matrix is computed as $\begin{array}{r}5\cdot 0+6\cdot 2+9\cdot 1=21\end{array}$ .
    ::新矩阵第二行第一列条目的计算为 50+62+91=21。
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  • The entry in the first row second column of the new matrix is computed as  $\begin{array}{r}1\cdot 1+4\cdot 0+3\cdot 1=4\end{array}$  
    ::新矩阵第一行第二列第二列条目的计算值为 11+40+31=4
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  • The entry in the second row second column of the new matrix is computed as  $\begin{array}{r}5\cdot 1+6\cdot 0+9\cdot 1=14\end{array}$  
    ::新矩阵第二行第二列第二列条目的计算值为 51+60+91=14。

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  Continue this pattern and you will find that the solution to this multiplication is: 
  

  $$\begin{array}{rl}C& =\left[\begin{array}{ccccc}11& 4& 12& 9& 7\\ 21& 14& 42& 17& 15\end{array}\right]\end{array}$$

  
  ::继续此模式, 你会发现此乘法的解决方案是: C=[11412972114421715]

             

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  Other Properties of Matrix Algebra
  ::矩阵代数的其他属性

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  • Commutativity holds for matrix addition . This means that when matrices  $\begin{array}{r}A\end{array}$ and  $\begin{array}{r}B\end{array}$ can be added (when they have matching orders), then: $\begin{array}{r}A+B=B+A\end{array}$
    ::即当可以添加矩阵A和B时(当它们有匹配的订单时),那么:A+B=B+A
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  • Commutativity does not hold in general for .
    ::通信性一般不能维持......。
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  • Associativity holds for both multiplication and addition. $\begin{array}{r}(AB)C=A(BC),(A+B)+C=A+(B+C)\end{array}$
    :AB)C=A(BC),(A+B)+C=A+(B+C)
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  • Distribution over addition and subtraction holds. $\begin{array}{r}A(B\pm C)=AB\pm AC\end{array}$
    ::A(BC)=ABAC

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked what the differences between matrix and regular algebra are. The main difference between matrix algebra and regular algebra with numbers is that matrices do not have the commutative property for multiplication.  There are other complexities that matrices have, but many of them stem from the fact that for most matrices $\begin{array}{r}AB\ne BA\end{array}$ . 
  ::早些时候,有人问您矩阵和常规代数之间的差别是什么。矩阵代数和具有数字的常规代数之间的主要区别在于矩阵不具备乘法的通量属性。 矩阵还存在其他复杂之处, 但其中许多复杂之处来自大多数矩阵ABBA。

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  Example 2
  ::例2

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  Show the commutative property does not hold by demonstrating $\begin{array}{r}AB\ne BA\end{array}$
  ::通过演示 ABBABA 显示通量属性不持有

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  $$\begin{array}{r}A=\left[\begin{array}{ccc}0& -1& 8\\ 1& 2& 0\\ 4& 3& 12\end{array}\right],B=\left[\begin{array}{ccc}1& 5& 1\\ 2& 2& 1\\ 4& 3& 0\end{array}\right]\end{array}$$

  
  ::A=[0-181204312],B=[151221430]

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  $$\begin{array}{rl}AB& =\left[\begin{array}{ccc}30& 22& -1\\ 5& 9& 3\\ 58& 62& 7\end{array}\right]\\ BA& =\left[\begin{array}{ccc}9& 12& 20\\ 6& 5& 28\\ 3& 2& 32\end{array}\right]\end{array}$$

  
  ::AB=[3022-159358627]BA=[9122065283232]

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  Example 3
  ::例3

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  Compute the following matrix arithmetic: $\begin{array}{r}10\cdot (2A-3C)\cdot B\end{array}$ .
  ::计算下列矩阵算术: 10(2A-3C)B。

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  $\begin{array}{r}A=\left[\begin{array}{cc}1& 2\\ 4& 5\end{array}\right],B=\left[\begin{array}{ccc}0& 1& 2\\ 4& 3& 2\end{array}\right],C=\left[\begin{array}{cc}12& 0\\ 1& 3\end{array}\right]\end{array}$
  ::A=[1245],B=[012432],C=[2013]

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  When a matrix is multiplied by a scalar (such as with $\begin{array}{r}2A\end{array}$ ), multiply each entry in the matrix by the scalar.
  ::当一个矩阵乘以一个标量(如2A)时,将矩阵中每个条目乘以标量。

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  $$\begin{array}{rl}2A& =\left[\begin{array}{cc}2& 4\\ 8& 10\end{array}\right]\\ -3C& =\left[\begin{array}{cc}-36& 0\\ -3& -9\end{array}\right]\\ 2A-3C& =\left[\begin{array}{cc}-34& 4\\ 5& 1\end{array}\right]\end{array}$$

  
  ::2A=[24810]-3C=[-360-3-3-9]2A-3C=[-344451]

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  Since the associative property holds, you can either distribute the ten or multiply by matrix  $\begin{array}{r}B\end{array}$ next.
  ::自关联财产持有以来,您可以分配10或乘以下个矩阵B。

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  $$\begin{array}{rl}(2A-3C)\cdot B& =\left[\begin{array}{ccc}16& -22& -60\\ 4& 8& 12\end{array}\right]\\ 10\cdot (2A-3C)\cdot B& =\left[\begin{array}{ccc}160& -220& -600\\ 40& 80& 120\end{array}\right]\end{array}$$

  
  :2A-3C)B=[16-22-604812]10(2A-3C)B=[160-220-6004080120]

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  Example 4
  ::例4

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  Use your calculator to input and compute the following matrix operations.
  ::使用您的计算器输入和计算下列矩阵操作。

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  $$\begin{array}{rl}& A=\left[\begin{array}{ccc}54& 65& 12\\ 235& 322& 167\\ 413& 512& 123\end{array}\right],B=\left[\begin{array}{ccc}163& 212& 466\\ 91& 221& 184\\ 42& 55& 42\end{array}\right]\\ & A^T\cdot B\cdot A-100A\end{array}$$

  
  ::A=[5465122335322167413512123],B=[6321246691221184425542]ATBA-100A

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  Most graphing calculators like the TI-84 can do operations on matrices. Find where you can enter matrices and enter the two matrices.
  ::象 TI-84 这样的大多数图形计算计算器可以在矩阵上操作。 查找您可以输入矩阵和输入两个矩阵的位置 。

  

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  Then type in the appropriate operation and see the result.  The TI-84 has a built in Transpose button.
  ::然后在适当的操作中键入并查看结果。 TI- 84 在一个转换按钮中创建了 。

  

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  The actual numbers on this guided practice are less important than the knowledge that your calculator can perform all of the matrix algebra demonstrated in this concept.  It is useful to fully know the capabilities of the tools at your disposal, but it should not replace knowing why the calculator does what it does. 
  ::这一指导做法上的实际数字比知道计算器能够进行这个概念所显示的所有矩阵代数要不重要,充分了解你可用的工具的能力是有益的,但不应取代知道计算器为什么做它的工作。

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  Example 5
  ::例5

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  Matrix multiplication can be used as a transformation in the coordinate system.  Consider the triangle with coordinates (0, 0) (1, 2) and (1, 0) the following matrix:
  ::矩阵乘法可用作坐标系统中的转换。如果三角形具有坐标(0,0)(1,2)和(1,0),则考虑下列矩阵:

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  $\begin{array}{r}\left[\begin{array}{cc}\cos {90}^{\circ }& \sin {90}^{\circ }\\ -\sin 90& \cos 90\end{array}\right]\end{array}$
  ::[cos909090990909090]

  

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  What does the new picture look like? 
  ::新照片长什么样?

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  The matrix simplifies to become:
  ::矩阵简化为:

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  $\begin{array}{r}\left[\begin{array}{cc}\cos {90}^{\circ }& \sin {90}^{\circ }\\ -\sin 90& \cos 90\end{array}\right]=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]\end{array}$
  ::[cos9090909090909090]=[01~10]

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  When applied to each point as a transformation, a new point is produced. Note that  $\begin{array}{r}\left[\begin{array}{cc}x& y\end{array}\right]\end{array}$ is a matrix representing each original point and  $\begin{array}{r}\left[\begin{array}{cc}{x}^{\mathrm{\prime }}& y^{\mathrm{\prime }}\end{array}\right]\end{array}$ is the new point.  The  $\begin{array}{r}{x}^{\mathrm{\prime }}\end{array}$ is read as “ $\begin{array}{r}x\end{array}$ prime” and is a common way to refer to a result after a transformation.
  ::当应用到作为转换的每个点时,将产生一个新的点。请注意, [xy] 是一个代表每个原始点的矩阵, [x_y_] 是新点。 x_ 被解读为“x print ” , 是一种常见的表示转换后结果的方法。

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  $$\begin{array}{rl}\left[\begin{array}{cc}x& y\end{array}\right]\cdot \left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]& =\left[\begin{array}{cc}{x}^{\mathrm{\prime }}& y^{\mathrm{\prime }}\end{array}\right]\\ \left[\begin{array}{cc}0& 0\end{array}\right]\cdot \left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]& =\left[\begin{array}{cc}0& 0\end{array}\right]\\ \left[\begin{array}{cc}1& 2\end{array}\right]\cdot \left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]& =\left[\begin{array}{cc}-2& 1\end{array}\right]\\ \left[\begin{array}{cc}1& 0\end{array}\right]\cdot \left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]& =\left[\begin{array}{cc}0& 1\end{array}\right]\end{array}$$

  
  ::[xy]_[01-10]=[x_Y_[00]_[01-10]=[00][12]_[01-10]=[-21][10]_[01-10]_[01-10]=[01]=[01]

  

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  Notice how the matrix transformation rotates graphs in a counterclockwise direction $\begin{array}{r}{90}^{\circ }\end{array}$ .
  ::注意矩阵转换如何在逆时针方向90 旋转图表 。

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  $\begin{array}{r}\left[\begin{array}{cc}x& y\end{array}\right]\left[\begin{array}{cc}\cos {90}^{\circ }& \sin {90}^{\circ }\\ -\sin 90& \cos 90\end{array}\right]=\left[\begin{array}{cc}-y& x\end{array}\right]\end{array}$
  ::[xy]=[-yx]

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  The matrix transformation applied in the following order will rotate a graph clockwise $\begin{array}{r}{90}^{\circ }\end{array}$ .
  ::按以下顺序应用的矩阵变换将旋转一个图表顺时针 90\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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  $\begin{array}{r}\left[\begin{array}{cc}\cos {90}^{\circ }& \sin {90}^{\circ }\\ -\sin 90& \cos 90\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}y\\ -x\end{array}\right]\end{array}$
  ::[cos 90 90 90 90 [xy] =[y-x]

  Summary

  播放段落 Matrix addition and subtraction can only be performed on matrices of the same order, by summing or subtracting the corresponding entries. ::矩阵添加和减法只能在相同顺序的矩阵上进行,方法是对相应条目进行缩写或减法。 播放段落 Matrix multiplication can be performed if the number of columns in the first matrix matches the number of rows in the second matrix, resulting in a matrix with the number of rows from the first matrix and the number of columns from the second matrix. ::如果第一个矩阵中的列数与第二个矩阵中的行数相符,就可以进行矩阵乘法,从而产生一个矩阵,其中含有第一个矩阵中的行数和第二个矩阵中的列数。 播放段落 Commutativity holds for matrix addition but does not hold for matrix multiplication. ::Comutativity 支持增加矩阵,但不支持矩阵乘法。 播放段落 Associativity holds for both matrix addition and multiplication. ::矩阵添加和乘法都具有关联性。 播放段落 Distribution holds for both matrix addition and subtraction. ::矩阵增加和减法的分布保持不变。

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  Review
  ::回顾

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  Do #1-#11 without your calculator.
  ::#1 -11没有你的计算器。

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  $\begin{array}{r}A=\left[\begin{array}{cc}2& 7\\ 3& 8\end{array}\right],B=\left[\begin{array}{ccc}0& 5& 1\\ 3& 4& 6\end{array}\right],C=\left[\begin{array}{cc}14& 6\\ 1& 2\end{array}\right],D=\left[\begin{array}{cc}5& 0\\ 1& 2\end{array}\right]\end{array}$
  ::A=[2738],B=[051346],C=[14612],D=[5012]

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  1. Find $\begin{array}{r}AC\end{array}$ . If not possible, explain.
  ::1. 寻找AC。 如果不可能,请解释。

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  2. Find $\begin{array}{r}BA\end{array}$ . If not possible, explain.
  ::2. 寻找BA。如果不可能,请解释。

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  3. Find $\begin{array}{r}CA\end{array}$ . If not possible, explain.
  ::3. 寻找CA。如果不可能,请解释。

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  4. Find $\begin{array}{r}4B^T\end{array}$ . If not possible, explain.
  ::4. 查找4BT。 如果不可能,请解释。

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  5. Find $\begin{array}{r}A+C\end{array}$ . If not possible, explain.
  ::5. 查找A+C。 如果不可能,请解释。

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  6. Find $\begin{array}{r}D-A\end{array}$ . If not possible, explain.
  ::6. 寻找D-A. 如果不可能,请解释。

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  7. Find $\begin{array}{r}2(A+C-D)\end{array}$ . If not possible, explain.
  ::7. 查找2(A+C-D)。 如果不可能,请解释。

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  8. Find $\begin{array}{r}(A+C)B\end{array}$ . If not possible, explain.
  ::8. 查找(A+C)B. 如果不可能,请解释。

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  9. Find $\begin{array}{r}B(A+C)\end{array}$ . If not possible, explain.
  ::9. 查找B(A+C)。 如果不可能,请解释。

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  10. Show that $\begin{array}{r}A(C+D)=AC+AD\end{array}$ .
  ::10. 显示A(C+D)=AC+AD。

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  11. Show that $\begin{array}{r}A(C-D)=AC-AD\end{array}$ .
  ::11. 显示A(C-D)=AC-AD。

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  Practice using your calculator for #12-#15.
  ::使用您的计算器练习 #12 -#15。

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  $\begin{array}{r}E=\left[\begin{array}{ccc}312& 59& 34\\ 342& 156& 189\\ 783& 23& 133\end{array}\right],F=\left[\begin{array}{ccc}33& 72& 21\\ 93& 41& 94\\ 62& 75& 72\end{array}\right],G=\left[\begin{array}{ccc}11& 735& 67\\ 93& 456& 2\\ 94& 34& 0\end{array}\right]\end{array}$
  ::E=[312593434215618978323133],F=[337221934194627572],G=[1173556793456294340]

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  12. Find $\begin{array}{r}E+F+G\end{array}$ .
  ::12. 寻找E+F+G。

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  13. Find $\begin{array}{r}2E\end{array}$ .
  ::13. 查找2E。

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  14. Find $\begin{array}{r}4F\end{array}$ .
  ::14. 发现4F。

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  15. Find $\begin{array}{r}(E+F)G\end{array}$ .
  ::15. 查找(E+F)G.

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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 。