Does unit determinant weights preserve norm? • r/MachineLearning

In a typical neural network, the series of matrix transformations applied to the vectors in the dataset perform rotation and scaling on the vectors. In order to stop the scaling transformation, the matrix would need to have unit determinant. Is it possible to apply certain constraint on the matrix such that the determinant always stays 1 even after optimization weight update is done? If scaling is not being done, then that means only rotation is getting applied. Does this imply that the norm of the input vectors will be preserved throughout the network computations?