Certainly! Let's walk through the steps to determine the value of the unknown element [tex]\( x \)[/tex] in the matrix:
[tex]\[
\begin{array}{cccc}
-4 & -6 & -9 & -6 \\
x & 7 & 4 & -4 \\
9 & 3 & 2 & 6 \\
3 & -4 & 8 & -6
\end{array}
\][/tex]
We need to find the value of [tex]\( x \)[/tex] such that the determinant of the matrix is zero.
1. Construct the Determinant Equation:
First, we need to construct the determinant equation for the 4x4 matrix. This determinant will be a polynomial equation in terms of [tex]\( x \)[/tex].
2. Setting up the Determinant:
To set this up, we write the determinant of the matrix as follows:
[tex]\[
\det\begin{vmatrix}
-4 & -6 & -9 & -6 \\
x & 7 & 4 & -4 \\
9 & 3 & 2 & 6 \\
3 & -4 & 8 & -6
\end{vmatrix} = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Solving for [tex]\( x \)[/tex] involves expanding this determinant, forming a polynomial equation, and solving for the values of [tex]\( x \)[/tex] for which the determinant is zero.
4. Coefficient Identification and Polynomial Equation:
The determinant calculation results in a polynomial equation. Each term in this polynomial represents a combination of the matrix elements. The coefficients for this polynomial are determined through algebraic expansion of the determinant.
5. Finding the Roots:
Once we have the polynomial equation, we solve it by finding its roots. These solutions represent the possible values of [tex]\( x \)[/tex] that satisfy the determinant being zero.
6. Identifying the Real Solution:
Among these roots, we need to identify the real value of [tex]\( x \)[/tex] since the determinants of matrices typically deal with real numbers.
Upon solving this polynomial equation for the variable [tex]\( x \)[/tex], we find that the value of [tex]\( x \)[/tex] that makes the determinant zero is approximately:
[tex]\[
x = -34.126126126126124
\][/tex]
Thus, the unknown element [tex]\( x \)[/tex] in the matrix is:
[tex]\[
\boxed{-34.126126126126124}
\][/tex]
