What is the determinant of the coefficient matrix of this system?
4x-3y=-8
8x-3y=12

The coefficient matrix is build with its rows representing each equation, and its columns representing each variable.

So, you may write the matrix as

[tex] \left[\begin{array}{cc}\text{x-coefficient, 1st equation}&\text{y-coefficient, 1st equation}\\\text{x-coefficient, 2nd equation}&\text{y-coefficient, 2nd equation} \end{array}\right] [/tex]

which means

[tex] \left[\begin{array}{cc}4&-3\\8&-3\end{array}\right] [/tex]

The determinant is computed subtracting diagonals:

[tex] \left | \left[ \begin{array}{cc}a&b\\c&d\end{array}\right]\right | = ad-bc [/tex]

So, we have

[tex] \left | \left[\begin{array}{cc}4&-3\\8&-3\end{array}\right] \right | = 4(-3) - 8(-3) = -4(-3) = 12 [/tex]