Question

What is the determinant of the coefficient matrix of the system

Answer 1

All of the z coefficients are zero. This means that the matrix will have a column full of zeroes:

$\left(\begin{array}{ccc}4&-6&0\\3&3&0\\2&-12&0\end{array}\right)$

And every time a matrix has a row/column full of zeroes, its determinant is zero.