To find the determinant of the coefficient matrix of the given system of equations, we follow these steps:
The system of equations is:
[tex]\[
\begin{cases}
4x - 6y + 0z = -7 \\
3x + 3y + 0z = -2 \\
2x - 12y + 0z = -1
\end{cases}
\][/tex]
The coefficient matrix for this system is:
[tex]\[
\begin{pmatrix}
4 & -6 & 0 \\
3 & 3 & 0 \\
2 & -12 & 0
\end{pmatrix}
\][/tex]
Next, we need to find the determinant of this 3x3 matrix. The general formula for the determinant of a 3x3 matrix [tex]\(\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}\)[/tex] is given by:
[tex]\[
\text{det} = a(ei - fh) - b(di - fg) + c(dh - eg)
\][/tex]
For our specific matrix, the calculation would be:
[tex]\[
\text{det} = 4(3 \cdot 0 - (-12) \cdot 0) - (-6)(3 \cdot 0 - 2 \cdot 0) + 0(3 \cdot (-12) - 2 \cdot 3)
\][/tex]
Simplifying each term:
- The first term is: [tex]\(4(0) = 0\)[/tex]
- The second term is: [tex]\(-6(0) = 0\)[/tex]
- The third term is: [tex]\(0\)[/tex]
Adding these up:
[tex]\[
\text{det} = 0 + 0 + 0 = 0
\][/tex]
Therefore, the determinant of the coefficient matrix is:
[tex]\[
\boxed{0}
\][/tex]
