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.