# Determinants and volume of parallelotopes

The absolute value of a 2 by 2 matrix determinant is the area of a corresponding parallelogram with the 2 row vectors as sides.

The absolute value of a 3 by 3 matrix determinant is the volume of a corresponding parallelepiped with the 3 row vectors as sides.

Can it be generalized to n-D? The absolute value of a n by n matrix determinant is the volume of a corresponding n-parallelotope?

-

Yes it can. In fact, as Katie Banks noted, a determinant is an intuitive way of thinking about volumes. To summarise her argument, if we consider the vectors as a matrix, switching two rows, multiplying one by a constant or adding a linear combination will have the same effect on the volume as on the determinate. We can use these operations to transform any n-parallelotope to cube and note that the determinate matches the signed volume here, so it will match it everywhere as well.

-