Applications of 4x4 Determinant & Representations of nxn Determinants

[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.

 

[pmacias]

The 4x4 determinant, like the 3x3 determinant, has many applications in mathematics and science. One of the main applications of a 4x4 determinant is in solving systems of linear equations. This is because a 4x4 determinant can be used to find the unknown variables in a system of four linear equations with four unknowns. This is especially useful in engineering and physics, where systems of equations are commonly used to model real-world problems.

Another important application of a 4x4 determinant is in calculating the volume of a parallelepiped in 4-dimensional space. This is because the absolute value of a 4x4 determinant represents the volume of a parallelpiped formed by four vectors in 4-dimensional space. This concept is also used in computer graphics and animation, where 4x4 determinants are used to transform and manipulate objects in 3D space.

In terms of representation, a 4x4 determinant can also represent the cross product of two 4-dimensional vectors. This is similar to how a 3x3 determinant represents the cross product of two 3-dimensional vectors. This representation is useful in physics, particularly in electromagnetism and quantum mechanics.

As for nxn determinants, they have a wide range of applications depending on the size and context of the problem. For example, a 5x5 determinant can be used in finding the inverse of a 5x5 matrix, while a 6x6 determinant can be used in solving systems of six linear equations with six unknowns. In general, nxn determinants are used in linear algebra to solve systems of equations, find the inverse of matrices, and calculate volumes and areas in higher dimensions.

In conclusion, 4x4 determinants have many practical applications in mathematics and science, such as solving systems of linear equations and calculating volumes in 4-dimensional space. They also have a useful representation in terms of cross products of 4-dimensional vectors. Additionally, nxn determinants have a wide range of applications depending on the size and context of the problem, making them a versatile tool in various fields of study.