What Does a Determinant of 1 in a Transformation Matrix Signify?

[ns5032]

Does it mean anything in particular about the transformation if the determinant of a transformation matrix is 1?

 

[HallsofIvy]

Yes, it does. That means the transformation does not change the length of a vector nor does it reverse the direction. It is, basically, a "rotation".

 

[Dick]

det=1 is not sufficient to show a transformation is a rotation, though the converse is true. Consider a matrix like [[1/2,0],[0,2]]. What is true is that the transformation doesn't change the volume of a region.

 

[HallsofIvy]

Thanks, Dick. You are, of course, right.