lorentz symmetry - Physical/geometrical interpretaion of the determinant of a matrix

Consider a matrix transformation $\mathbf{T}$ that acts on a vector $\mathbf{x}$:

$$ \mathbf{x}' = \mathbf{T}\mathbf{x} $$.

Now, I know that one-dimensional linear transformations expand the length by a factor $|det(\mathbf{T})|$, two-dimensional linear transformations expand the area by a factor $|det(\mathbf{T})|$, and three-dimensional linear transformations expand the volume by a factor $|det(\mathbf{T})|$.

How can I see this mathematically though? I was thinking of calculating the norm $||\mathbf{x}'|| = ||\mathbf{T}||\cdot||\mathbf{x}||$ but don't know how to deal with $||\mathbf{T}||$

The physics of this question is: do all Lorentz transformations have determinant equal to 1? Because they preserve the space-time interval?

EDIT:

What I was asking was more about: how can I see that the determinant of a matrix carries information about the area/volume change of the system it is acting on? Mathematically, how can I show that the area spanned by two vectors is unchanged IFF the determinant of the transformation matrix is 1?

Answer

do all Lorentz transformations have determinant equal to 1? Because they preserve the space-time interval?

Yes, they do, but preservation of the interval is not the right way to think about the "why."

We have two facts:

1. 
   

   Lorentz transformations have Jacobian determinant 1.

   
   
   


2. 
   

   Lorentz transformations preserve the spacetime interval.

   
   

There is no close or simple relationship between these facts. In particular, 1 does not imply 2. For example, the Galilean transformations have Jacobian determinant 1, but they do not preserve the spacetime interval. Similarly, we could define a rotation in the $x-t$ plane, and it would have Jacobian determinant 1 but not preserve the spacetime interval.

It is true that 2 implies 1, but this is simply because 2 is sufficient to completely characterize the Lorentz transformations and therefore give all their properties indirectly.

For a general proof of the unit Jacobian that applies to both Galilean transformations and Lorentz transformations, see the answer to this question: Motivation for preservation of spacetime volume by Lorentz transformation?