What do 4x4 and nxn determinants represent and compute?

[thefreeman]

What are the applications of a 4x4 determinant and what can it represent? To make it more clear, a 3x3 determinant represents vector perpendicular to two vectors and a 3x3 determinant can also be used to calculate torque. So, what can a 4x4 determinant do? Also, if there is a nxn determinant that also does something, I'd be interested in knowing what that represents too.

 

[jedishrfu]

do you mean matrices? a 3x3 matrix can be used to rotate a vector in 3-space

a 4x4 matrix ican be used to scale, rotate and translate a vector in 3 space

http://www.euclideanspace.com/maths/geometry/affine/matrix4x4/

http://inside.mines.edu/~gmurray/ArbitraryAxisRotation/

 

[AlephZero]

Also, if there is a nxn determinant that also does something, I'd be interested in knowing what that represents too.

Find out about eigenvalues and eigenvectors, and how they relate to determinants.

 

[UltrafastPED]

Determinants are invariant under similarity transformations - often appearing as changes in coordinate systems - and have other useful properties related to matrix multiplication and inversion.

In a sense the determinant calculates a volume related to the region bounded by the column (or row) vectors. This works in all dimensions.