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.