Sec1Sess6

det (A, B, C) = <math>\begin{bmatrix}a1\ a2\ a3 ; b1\ b2\ b3; c1\ c2\ c3\end{bmatrix} = a1 \begin{bmatrix}b2\ b3 ; c2\ c3\end{bmatrix} - a2 \begin{bmatrix}b1\ b3 ; c1\ c3\end{bmatrix} + a3 \begin{bmatrix}b1\ b2 ; c1\ c2\end{bmatrix}</math>
method of calc det is per col and row of an index eliminate that row and col index and take determinant of what is left. minus is in middle calc for rotation of coords reason.

parallelepiped is 3d box with sides of parallelogram, can have volume found from vectors shown above

laplace expansion to find determinant: <math>det(A) = a ( ei - fh ) - b ( di - fg ) + c ( dh - eg ) </math> where <math>A = \langle \langle a b c \rangle \langle d e f \rangle \langle g h i \rangle \rangle </math>

lots of further notation and methods:
http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-6-volumes-and-determinants-in-space/MIT18_02SC_MNotes_d1.pdf
I did not want to rewrite here many of the content written in that doc

In laplace transform note a - b + c format of the row or column that the determinants are multiplied by. This is determined by the "checkerboard pattern".

volume can be found using vectors in arbitrary (any) numbers of dimensions with recursive determinants calculations using Laplase transform or other volume finding methods.