Deriving lorentz transformation equations using linear transformation

Question

While I was deriving the Lorentz transformation equations using linear transformation and eigenvectors i noticed that i get the correct equation only when determinant of the transformation matrix is $\pm 1$.What is the physical meaning behind this result?

Answer 1

The determinant of a transformation matrix is generally interpreted as the volume scaling factor of the transformation, but this does not give a direct physical interpretation for the determinant being $\pm 1$. It results from the fact that the inverse of a Lorentz boost $v$ is a Lorentz boost $-v$, and that these boosts are identical in magnitude, so must have the same determinant, and that transformations of space coordinates (e.g. space rotation) make no scaling changes.