Calculate the volume of a tetrahedron enclosed by the following 4 planes
$$a_1x+b_1y+c_1z+d_1=0\\a_2x+b_2y+c_2z+d_2=0\\a_3x+b_3y+c_3z+d_3=0\\a_4x+b_4y+c_4z+d_4=0$$
The easiest way to calculate this volume is by applying a linear transformation to the tetrahedron where the planes of the tetrahedron align to the coordinate planes. It is impossible to find a purely linear transformation that does this, as all linear transformations project the origin, to the origin, but it is possible to find a linear transformation, followed by a translation, to do this. As translations do not change the volume, this will not pose any unsolvable issues.
An example of such a transformation is
$$u=a_1x+b_1y+c_1z+d_1\\v=a_2x+b_2y+c_2z+d_2\\w=a_3x+b_3y+c_3z+d_3$$
To find the transformation of a given plane, one has to add the equation of the plane to the system of equations of the transformation. For example, for the first plane this becomes:
$$u=a_1x+b_1y+c_1z+d_1\\v=a_2x+b_2y+c_2z+d_2\\w=a_3x+b_3y+c_3z+d_3\\0=a_1x+b_1y+c_1z+d_1$$
Or solved: $u=0$. So the first plane matches the $vw$-plane.
This is analog for the two next planes, but things become interesting for the last plane, where the system becomes:
$$u=a_1x+b_1y+c_1z+d_1\\v=a_2x+b_2y+c_2z+d_2\\w=a_3x+b_3y+c_3z+d_3\\0=a_4x+b_4y+c_4z+d_4$$
Now, in our $uvw$ space, we have a tetrahedron enclosed by the coordinate planes, and a 4th plane that is the solution of the above equation. Tetrahedra enclosed by all 3 coordinate planes have the following vertices: $$(0,0,0), (u,0,0), (0,v,0), (0,0,w)$$ Each of these points can be found by solving the system of equations of 3 of the 4 planes.
Using the first 3 planes obviously gives (0,0,0) as a solution.
Using all of them except the first gives us:
$$u=a_1x+b_1y+c_1z+d_1\\v=a_2x+b_2y+c_2z+d_2\\w=a_3x+b_3y+c_3z+d_3\\0=a_4x+b_4y+c_4z+d_4\\v=0\\w=0$$
This easily simplifies to
$$u=a_1x+b_1y+c_1z+d_1\\0=a_2x+b_2y+c_2z+d_2\\0=a_3x+b_3y+c_3z+d_3\\0=a_4x+b_4y+c_4z+d_4$$
If we call:
$$A_1 =\left[ \begin{matrix} a_1&b_1&c_1&1\\a_2&b_2&c_2&0\\a_3&b_3&c_3&0\\a_4&b_4&c_4&0\end{matrix}\right]\\A_2 =\left[ \begin{matrix} a_1&b_1&c_1&0\\a_2&b_2&c_2&1\\a_3&b_3&c_3&0\\a_4&b_4&c_4&0\end{matrix}\right]\\A_3 =\left[ \begin{matrix} a_1&b_1&c_1&0\\a_2&b_2&c_2&0\\a_3&b_3&c_3&1\\a_4&b_4&c_4&0\end{matrix}\right]\\A_4 =\left[ \begin{matrix} a_1&b_1&c_1&0\\a_2&b_2&c_2&0\\a_3&b_3&c_3&0\\a_4&b_4&c_4&1\end{matrix}\right]$$
$$D = \left[\begin{matrix} d_1\\d_2\\d_3\\d_4 \end{matrix}\right]$$
$$X = \left[\begin{matrix} x\\y\\z\\-u \end{matrix}\right]$$
The system can be rewritten as
$$A_1X+D=0$$
$$X=-A_1^{-1}D $$
Now as
$$u = -\left[\begin{matrix} 0&0&0&1 \end{matrix}\right]X$$
$$u = \left[\begin{matrix} 0&0&0&1 \end{matrix}\right]A_1^{-1}D$$
$$u = \frac{1}{\det{A_1}}\left[\begin{matrix} 0&0&0&1 \end{matrix}\right]\text{adj}{A_1}D$$
As the bottom row of $\text{adj}{A_1}$ equals
$$\left| \begin{matrix}a_2&b_2&c_2\\a_3&b_3&c_3\\a_4&b_4&c_4\end{matrix}\right|;\left| \begin{matrix} a_1&b_1&c_1\\a_3&b_3&c_3\\a_4&b_4&c_4\end{matrix}\right|;\left| \begin{matrix} a_1&b_1&c_1\\a_2&b_2&c_2\\a_4&b_4&c_4\end{matrix}\right|;\left| \begin{matrix} a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{matrix}\right|$$
Then
$$\left[\begin{matrix} 0&0&0&1 \end{matrix}\right]\text{adj}{A_1}D = d_1\det{A_1}+d_2\det{A_2}+d_3\det{A_3}+d_4\det{A_4}$$
If one defines:
$$M =\left[ \begin{matrix} a_1&b_1&c_1&d_1\\a_2&b_2&c_2&d_2\\a_3&b_3&c_3&d_3\\a_4&b_4&c_4&d_4\end{matrix}\right]$$
Then is
$$u = \frac{\det{M}}{\det{A_1}}$$
Analog is
$$v = \frac{\det{M}}{\det{A_2}}\\w =\frac{\det{M}}{\det{A_3}}$$
Now the volume of the tetrahedron is the surface area of one of the faces time the height divided by 3, so if one takes the face to be the $vw$-plane, the volume is
$$\frac{u}{3} \times \frac{vw}{2} = \frac{uvw}{6}$$
In $uvw$ space. In $xyz$ space the volume has to be divided by the Jacobian:
$$\left| \begin{matrix} a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{matrix}\right|$$
which is $\det{A_4}$ so the final volume becomes:
$$\frac{\det{M}^3}{6\det{A_1}\det{A_2}\det{A_3}\det{A_4}}$$
Or expanded:
$$\frac{{\left| \begin{matrix} a_1&b_1&c_1&d_1\\a_2&b_2&c_2&d_2\\a_3&b_3&c_3&d_3\\a_4&b_4&c_4&d_4\end{matrix}\right|^3}}{6\left| \begin{matrix}a_2&b_2&c_2\\a_3&b_3&c_3\\a_4&b_4&c_4\end{matrix}\right|\left| \begin{matrix} a_1&b_1&c_1\\a_3&b_3&c_3\\a_4&b_4&c_4\end{matrix}\right|\left| \begin{matrix} a_1&b_1&c_1\\a_2&b_2&c_2\\a_4&b_4&c_4\end{matrix}\right|\left| \begin{matrix} a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{matrix}\right|}$$