$4\times 4$ Matrix determinant (For computer graphics)

Question

So Opengl and other graphics Api's use Matrices that are $4\times 4$, because they have to include affine transformations (translation). The 4th row and column are included for this reason. The determinant can be used to find the inverse of a transformation. So my big question is do you find the determinant of the $3\times 3$ Matrix that represents the linear transformations or the whole $4\times 4$ Matrix? (I am aware translations have inverses even though they are not linear).

Answer 1

i think you need the determinant of the entire matrix. But to find the inverse, you rarely use the determinant -- instead, you solve (using Gaussian Elimination) the equation $A\vec{x} = \vec{b}$ for $\vec{x}$ with $A, \vec{b}$ given - it is a more stable numerical approach than actually computing $A^{-1}$...