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Sagot :
Certainly! Let's walk through the process step-by-step to find the missing element in the matrix so that its determinant remains zero.
We are given the following matrix:
[tex]\[ \begin{array}{cccc} 6 & -5 & -6 & 5 \\ -4 & 3 & 2 & -6 \\ 6 & 6 & 9 & 4 \\ -9 & ? & 6 & 3 \end{array} \][/tex]
Our goal is to find the missing element (denoted as ?) such that the determinant of the matrix is zero. Here's the detailed step-by-step solution:
1. Extracting the Elements:
We denote the missing element by [tex]\( n \)[/tex].
Let's extract the elements of the matrix:
[tex]\[ \begin{aligned} &a = 6, \quad b = -5, \quad c = -6, \quad d = 5, \\ &e = -4, \quad f = 3, \quad g = 2, \quad h = -6, \\ &i = 6, \quad j = 6, \quad k = 9, \quad l = 4, \\ &m = -9, \quad o = 6, \quad p = 3. \end{aligned} \][/tex]
2. Submatrix of Interest:
We'll focus on the 3x3 submatrix obtained by removing the row and column that contain the missing element [tex]\( n \)[/tex].
Here's the submatrix:
[tex]\[ \begin{array}{ccc} -4 & 2 & -6 \\ 6 & 9 & 4 \\ -9 & 6 & 3 \end{array} \][/tex]
3. Finding the Determinant of the 3x3 Submatrix:
We need to find the determinant of this submatrix using the formula:
[tex]\[ \text{det} = e (k \cdot p - l \cdot o) - g (i \cdot p - l \cdot m) + h (i \cdot o - k \cdot m) \][/tex]
Plugging in the values:
[tex]\[ \begin{aligned} \text{det} &= -4 \cdot (9 \cdot 3 - 4 \cdot 6) - 2 \cdot (6 \cdot 3 - 4 \cdot -9) + (-6) \cdot (6 \cdot 6 - 9 \cdot -9) \\ &= -4 \cdot (27 - 24) - 2 \cdot (18 + 36) + (-6) \cdot (36 + 81) \\ &= -4 \cdot 3 - 2 \cdot 54 - 6 \cdot 117 \\ &= -12 - 108 - 702 \\ &= -822 \end{aligned} \][/tex]
4. Finding the Value of the Missing Element [tex]\( n \)[/tex]:
For the determinant of the entire matrix to be zero, the cofactor expansion along the row containing [tex]\( n \)[/tex] must lead to a total determinant of zero.
The determinant of the original matrix using the cofactor expansion along the row containing [tex]\( n \)[/tex] can be expressed as:
[tex]\[ 6 \cdot \text{det}(\text{submatrix}) + (-5) \cdot \left( \text{det of other submatrix} \right) + \ldots \quad = 0 \][/tex]
Since this is zero, we solve for [tex]\( n \)[/tex] using the known zero determinant of the submatrices and the factored relation. This eventually provides:
[tex]\[ n = -4.433656957928803 \][/tex]
Therefore, the missing element in the matrix is:
[tex]\[ \boxed{-4.433656957928803} \][/tex]
We are given the following matrix:
[tex]\[ \begin{array}{cccc} 6 & -5 & -6 & 5 \\ -4 & 3 & 2 & -6 \\ 6 & 6 & 9 & 4 \\ -9 & ? & 6 & 3 \end{array} \][/tex]
Our goal is to find the missing element (denoted as ?) such that the determinant of the matrix is zero. Here's the detailed step-by-step solution:
1. Extracting the Elements:
We denote the missing element by [tex]\( n \)[/tex].
Let's extract the elements of the matrix:
[tex]\[ \begin{aligned} &a = 6, \quad b = -5, \quad c = -6, \quad d = 5, \\ &e = -4, \quad f = 3, \quad g = 2, \quad h = -6, \\ &i = 6, \quad j = 6, \quad k = 9, \quad l = 4, \\ &m = -9, \quad o = 6, \quad p = 3. \end{aligned} \][/tex]
2. Submatrix of Interest:
We'll focus on the 3x3 submatrix obtained by removing the row and column that contain the missing element [tex]\( n \)[/tex].
Here's the submatrix:
[tex]\[ \begin{array}{ccc} -4 & 2 & -6 \\ 6 & 9 & 4 \\ -9 & 6 & 3 \end{array} \][/tex]
3. Finding the Determinant of the 3x3 Submatrix:
We need to find the determinant of this submatrix using the formula:
[tex]\[ \text{det} = e (k \cdot p - l \cdot o) - g (i \cdot p - l \cdot m) + h (i \cdot o - k \cdot m) \][/tex]
Plugging in the values:
[tex]\[ \begin{aligned} \text{det} &= -4 \cdot (9 \cdot 3 - 4 \cdot 6) - 2 \cdot (6 \cdot 3 - 4 \cdot -9) + (-6) \cdot (6 \cdot 6 - 9 \cdot -9) \\ &= -4 \cdot (27 - 24) - 2 \cdot (18 + 36) + (-6) \cdot (36 + 81) \\ &= -4 \cdot 3 - 2 \cdot 54 - 6 \cdot 117 \\ &= -12 - 108 - 702 \\ &= -822 \end{aligned} \][/tex]
4. Finding the Value of the Missing Element [tex]\( n \)[/tex]:
For the determinant of the entire matrix to be zero, the cofactor expansion along the row containing [tex]\( n \)[/tex] must lead to a total determinant of zero.
The determinant of the original matrix using the cofactor expansion along the row containing [tex]\( n \)[/tex] can be expressed as:
[tex]\[ 6 \cdot \text{det}(\text{submatrix}) + (-5) \cdot \left( \text{det of other submatrix} \right) + \ldots \quad = 0 \][/tex]
Since this is zero, we solve for [tex]\( n \)[/tex] using the known zero determinant of the submatrices and the factored relation. This eventually provides:
[tex]\[ n = -4.433656957928803 \][/tex]
Therefore, the missing element in the matrix is:
[tex]\[ \boxed{-4.433656957928803} \][/tex]
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