Notation: For an matrix by we mean the submatrix of which is obtained by deleting the row and column.
The proof of the next theorem is omitted. The interested reader is advised to go through Appendix 14.3.
Recall that the dot product, and is the length of the vector We denote the length by With the above notation, if is the angle between the vectors and then
Which tells us,
Note here that if then
Let be the parallelopiped formed with as a vertex and the vectors as adjacent vertices. Then observe that is a vector perpendicular to the plane that contains the parallelogram formed by the vectors and So, to compute the volume of the parallelopiped we need to look at where is the angle between the vector and the normal vector to the parallelogram formed by and So,
Hence,
In general, for any matrix it can be proved that is indeed equal to the volume of the -dimensional parallelopiped. The actual proof is beyond the scope of this book.