Is Determinant Multiplication Commutative and What Does |AB|=|BA| Imply?

[phymatter]

is multiplication of determinant commutative ?
if so , then is this correct: |A||B|=|B||A|=|AB|=|BA| ?
then what does |AB|=|BA| imply ??

 

[tiny-tim]

hi phymatter!

is multiplication of determinant commutative ?
if so , then is this correct: |A||B|=|B||A|=|AB|=|BA| ?

yup!

then what does |AB|=|BA| imply ??

that all is right with the world!

 

[chiro]

is multiplication of determinant commutative ?
if so , then is this correct: |A||B|=|B||A|=|AB|=|BA| ?
then what does |AB|=|BA| imply ??

The determinant has a lot of different uses.

In one sense it measures a linear measure in some dimension (in general we call it a parallelpiped in dimension n).

Another common use is to test linear dependence of a set of vectors (unit basis or elsewise).

Typically when we think about the product of two matrices, one way to view them is to view them as a composition of maps (or composition of linear functions).

Like if we had a function f(g(x)) using a AB we have f(g(A)) where A isn't necessarily a single number or even a vector, but instead a general matrix.

So this measure of a transformation gives information to how it changes another system (usually a vector but sometimes a matrix).

For example a rotation matrix has a determinant of 1. This means that the transformation preserves length when applied to some vector input. So if you had two rotation matrices and you used composition of maps (in this case ABC where they are all rotation matrices), then the composition of all these maps applied to any vector will conserve the length and computationally this can be proved using det(ABC) = det(A) x det(B) x det(C) = 1 x 1 x 1 = 1. This is a good property to have since any composition of rotations will preserve length.

In physics, you will have things called groups and a lot of groups used in high level physics have certain properties like the rotation matrices I mentioned above. Things like Special Unitary (SU(N)) have this kind of property.

 

[phymatter]

thanks tiny-tim and chiro!