4.7 寻找儿科决定因素

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  寻找母体决定因素

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  A mathematical theorem states that a matrix is singular if and only if its determinant is zero. Is the following matrix singular?
  ::数学定理指出,矩阵是单数的,只有其决定因素为零时才如此。下面的矩阵是单数的吗?

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

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  Determinant of a Matrix
  ::矩阵的决定因素

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  Each square matrix has a real number value associated with it called its determinant. This value is denoted by $\begin{array}{r}detA\end{array}$ or $\begin{array}{r}|A|\end{array}$ .
  ::每个平方矩阵都有与其称为决定因素的实际数字值相关的实际数字值,该数值用“A”或“A”表示。

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  Finding the Determinant of a $\begin{array}{r}2\times 2\end{array}$ matrix:
  ::查找 2x2 矩阵的决定因素 :

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  $$\begin{array}{r}det\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc\end{array}$$

  
  ::[abcd] abcdad-bc

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  Finding the Determinant of a $\begin{array}{r}3\times 3\end{array}$ matrix: To begin, we will repeat the first two columns after matrix. Next, calculate the products and sums as shown below and find the difference . The result is the determinant of the $\begin{array}{r}3\times 3\end{array}$ matrix.
  ::查找 3x3 矩阵的决定因素 : 要开始, 我们将在矩阵之后重复前两列 。 接下来, 计算下文所示的产品和总和, 并找出差异 。 结果是 3x3 矩阵的决定因素 。

  

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  Using the determinant to find the Area of a Triangle in the coordinate plane :
  ::使用决定因素在坐标平面上找到三角区域:

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  We can find the area of a triangle with vertices $\begin{array}{r}(x_1,y_1),(x_2,y_2)\end{array}$ and $\begin{array}{r}(x_3,y_3)\end{array}$ using the formula below
  ::使用以下公式,我们可以找到带有顶部(x1,y1),(x2,y2)和(x3,y3)的三角形区域。

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  $$\begin{array}{r}A=\pm \frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|,\end{array}$$

  where the $\begin{array}{r}\pm \end{array}$ accounts for the possibility that the determinant could be negative but area should always be positive.
  ::A+12_x1y11x2y2y21x3y31{},其中计算出的决定因素可能是负的,但面积应始终是正的。

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  Using the calculator to find the determinant of a matrix: 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. Now return to the home screen ( $\begin{array}{r}{2}^{nd}\end{array}$ MODE QUIT) and return to the MATRIX menu. Arrow over to MATH and select 1:det( by pressing ENTER. Go into the MATRIX menu once more to select 1: $\begin{array}{r}[A]\end{array}$ under the NAMES column. Press ENTER. Your screen should show $\begin{array}{r}det([A]\end{array}$ at this time. Press ENTER once more and the result will be your determinant. These directions work for square matrices of any size.
  ::使用计算器找到矩阵的决定因素 : 如果您正在使用 TI- 83 或 TI- 84 , 请按 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 。 现在返回家用屏幕 (2nd MODE QUIT) 并返回 MATRIX 菜单 。 箭头转到 MATH 并选择 1: ditt (按 ENTER 键 ) 。 再次进入 MATRIX 菜单, 在 NAMES 列下选择 1: [A] 。 按 ENTER 。 您的屏幕此时应该显示 det ([A] 。 按 以后再按 ENTER 键, 其结果将是您的决定因素 。 这些方向对任意的平方矩阵都起作用 。

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  Let's solve the following problems relating to determinants of matrices.
  ::让我们解决与矩阵决定因素有关的下列问题。

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  1. Find the
     $$\begin{array}{r}det\left[\begin{array}{cc}3& -4\\ 1& 5\end{array}\right].\end{array}$$
     
     ::找到该裁决[3-415]。

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  Using the rule above for a $\begin{array}{r}2\times 2\end{array}$ matrix, the determinant can be found as shown:
  ::使用上述规则对 2xx2 矩阵使用2x2 矩阵,可以找到决定因素如下:

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  $$\begin{array}{r}det\left[\begin{array}{cc}3& -4\\ 1& 5\end{array}\right]=\left|\begin{array}{cc}3& -4\\ 1& 5\end{array}\right|=(3)(5)-(-4)(1)=15-(-4)=19\end{array}$$

  
  ::[3-415]________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  2. Find the
     $$\begin{array}{r}det\left[\begin{array}{ccc}2& -3& 5\\ -4& 7& 1\\ 3& 8& 6\end{array}\right].\end{array}$$
     
     ::查找该裁决[2-35-47-1386]。

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  First, we need to repeat the first two columns. Then we can find the diagonal products as shown:
  ::首先,我们需要重复前两栏。然后我们可以找到对角产品,如:

  $$\begin{array}{r}\left[\begin{array}{ccc}2& -3& 5\\ -4& 7& 1\\ 3& 8& 6\end{array}\right]\begin{array}{cc}2& -3\\ -4& 7\\ 3& 8\end{array}\\ (2\cdot 7\cdot 6)+(-3\cdot 1\cdot 3)+(5\cdot -4\cdot 8)& =84+-9+-160=-85\\ =(3\cdot 7\cdot 5)+(8\cdot 1\cdot 2)+(6\cdot -4\cdot -3)& =105+16+72=193\\ -85-193& =-278\end{array}$$

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  3. Find the area of the triangle with vertices (2, -1), (4, 5) and (8, 1)
     ::查找三角形的三角形区域,以顶点(2,-1),(4,5)和(8,1)

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  The first step is to set up our matrix and find the determinant as shown:
  ::第一步是建立我们的矩阵,并找到以下的决定因素:

  $$\begin{array}{r}\left[\begin{array}{ccc}2& -1& 1\\ 4& 5& 1\\ 8& 1& 1\end{array}\right]\begin{array}{cc}2& -1\\ 4& 5\\ 8& 1\end{array}\\ (2\cdot 5\cdot 1)+(-1\cdot 1\cdot 8)+(1\cdot 4\cdot 1)& =10+-8+4=6\\ =(8\cdot 5\cdot 1)+(1\cdot 1\cdot 2)+(1\cdot 4\cdot -1)& =40+2-4=38\\ 6-38& =-32\end{array}$$

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  Now we can multiply this determinant, -32, by $\begin{array}{r}-\frac{1}{2}\end{array}$ (we will multiply by the negative in order to have a positive result) to get 16. So the area of the triangle is 16 $\begin{array}{r}{u}^2\end{array}$ .
  ::现在我们可以乘以这个决定因素 - 32 乘以 - 12 (我们将以负乘以负乘以正结果) 获得16。 所以三角形的面积是 16 u2。

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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 whether the following matrix is singular. 
  ::早些时候,有人要求你确定以下矩阵是否单数。

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

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  To find the determinant, we first need to repeat the first two columns. Then we can find the diagonal products as shown:
  ::为了找到决定性因素,我们首先需要重复前两栏。然后我们就可以找到对角产品,如:

  $$\begin{array}{r}\left[\begin{array}{ccc}2& 1& 3\\ 0& 2& 1\\ -1& 3& 0\end{array}\right]\begin{array}{cc}2& 1\\ 0& 2\\ -1& 3\end{array}\\ (2\cdot 2\cdot 0)+(1\cdot 1\cdot -1)+(3\cdot 0\cdot 3)& =0+-1+0=-1\\ =(3\cdot 2\cdot -1)+(2\cdot 1\cdot 3)+(1\cdot 0\cdot 0)& =-6+6+0=0\\ -1-0& =-1\end{array}$$

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  The determinant is not zero and therefore the matrix is not singular.
  ::决定因素不是零,因此矩阵并非单一。

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  Example 2
  ::例2

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  Find the determinant of the matrix below:
  ::确定矩阵的决定因素如下:

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

  
  

  $$\begin{array}{r}\left|\begin{array}{cc}-1& 8\\ 2& -9\end{array}\right|=(-1)(-9)-(8)(2)=9-16=-7\end{array}$$

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  Example 3
  ::例3

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  Find the determinant of the matrix below:
  

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

  
  ::确定以下矩阵的决定因素:[-24-35-31-61-41-2]

  
  

  $$\begin{array}{r}\left|\begin{array}{ccc}-2& 4& -3\\ 5& -6& 1\\ -4& 1& -2\end{array}\right|\begin{array}{cc}-2& 4\\ 5& -6\\ -4& 1\end{array}\\ (-2\cdot -6\cdot -2)+(4\cdot -1\cdot -4)+(-3\cdot 5\cdot 1)& =-24-16-15=-55\\ =(-4\cdot -6\cdot -3)+(1\cdot 1\cdot -2)+(-2\cdot 5\cdot 4)& =-72-2-40=-114\\ -55-(-114)& =59\end{array}$$

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  Example 4
  ::例4

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  Find the area of the triangle with vertices (-5, 2), (8, -1) and (3, 9).
  ::找到三角形的三角形区域,有顶部( 5, 2) 、 (8, 1) 和 (3, 9) 。

  $$\begin{array}{r}\left|\begin{array}{ccc}-5& 2& 1\\ 8& -1& 1\\ 3& 9& 1\end{array}\right|\begin{array}{cc}-5& 2\\ 8& -1\\ 3& 9\end{array}\\ =(5+6+72)-(-3+-45+16)=83-(-32)=115\end{array}$$

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  So the area is $\begin{array}{r}\frac{1}{2}(115)=57.5u^2\end{array}$ .
  ::区域是12(115)=57.5 u2。

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  Review
  ::回顾

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  Find the determinants of the matrices. Use your calculator to check your answers.
  ::查找矩阵的决定因素。 使用您的计算器检查您的答案 。

  1. .

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

  2. .

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

  3. .

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

  4. .

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

  5. .

  $$\begin{array}{r}\left[\begin{array}{cc}11& 3\\ 7& 2\end{array}\right]\end{array}$$

  6. .

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

  7. .

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

  8. .

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

  9. .

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

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  Find the area of each triangle with vertices given below.
  ::查找每个三角形的以下顶点区域。

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  10. (2, -1), (-5, 2) and (0, 6)
      :2,1, (5, 2) 和 (0, 6))
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  11. (-8, 12), (10, 5) and (1, -4)
      :8, 12), (10, 5)和(1, 4)
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  12. (-7, 2), (8, 0) and (3, -4)
      :7,2),(8,0)和(3,4)

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  Find the value of $\begin{array}{r}a\end{array}$ in the matrices below.
  ::在下面的矩阵中查找一个值。

  13. .

  $$\begin{array}{r}\left|\begin{array}{cc}a& 3\\ 8& 2\end{array}\right|=-10\end{array}$$

  14. .

  $$\begin{array}{r}\left|\begin{array}{cc}4& a\\ 3& 5\end{array}\right|=-1\end{array}$$

  15. .

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  $$\begin{array}{r}\left|\begin{array}{ccc}2& -1& 3\\ 4& 5& 2\\ -3& 0& a\end{array}\right|=23\end{array}$$

  
  ::2 - 13452 - 30a 23

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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 。