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]
