To find the determinant of the coefficient matrix of the given system, we need to consider the matrix formed by the coefficients of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex] from each of the equations in the system. The system of linear equations is:
[tex]\[
\begin{cases}
4x - 6y + 0z = -7 \\
3x + 3y + 0z = -2 \\
2x - 12y + 0z = -1
\end{cases}
\][/tex]
The coefficient matrix [tex]\(A\)[/tex] corresponding to this system is:
[tex]\[
A = \begin{pmatrix}
4 & -6 & 0 \\
3 & 3 & 0 \\
2 & -12 & 0
\end{pmatrix}
\][/tex]
Now, we need to find the determinant of this 3x3 matrix. Recall the formula for the determinant of a 3x3 matrix [tex]\( A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \)[/tex]:
[tex]\[
\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\][/tex]
Applying this formula to our coefficient matrix, we have:
[tex]\[
A = \begin{pmatrix}
4 & -6 & 0 \\
3 & 3 & 0 \\
2 & -12 & 0
\end{pmatrix}
\][/tex]
We can see that the third column of the matrix consists entirely of zeros. When a matrix has a column consisting entirely of zeros, its determinant is always zero because any combination of submatrices used to calculate the determinant will contain this column of zeros, thus nullifying the product.
Therefore, the determinant of the given coefficient matrix is:
[tex]\[
\boxed{0}
\][/tex]
