Matrices With Determinant 1 Do Not Preserve Length
It is a necessary requirement for a vector to have the same length or magnitude after a linear transformation that the matrix associated with transformation have determinant 1, but this is not sufficient. The columns of thew matrix should also have magnitude 1, and should be orthogonal to each other.For example, let a linear transformation have associated matrix
\[ \left( \begin{array}{cc} 2 & 1 \\ 5 & 3 \end{array} \right) \]
.This matrix has determinant
\[2 \times 3 - 1 \times 5=1\]
The length of
\[ \begin{pmatrix}1\\0\end{pmatrix}\]
is 1 but \[ \left( \begin{array}{cc} 2 & 1 \\ 5 & 3 \end{array} \right) \begin{pmatrix}1\\0\end{pmatrix} = \begin{pmatrix}2\\5\end{pmatrix} \]
.The length of this vector is definitely not 1.
All length preserving matrices are in fact rotations or transformations or some sequence of these. Rotating or reflecting a vector, or any sequences of these does not change the length of the vector.