I was doing some algebra exercise and I noticed that to normalize an orthogonal matrix $M_{n \times n}$, I can just divide the matrix using the nᵗʰ root of the absolute value of the determinant.
I know that the determinant can be defined as how much the size of the area, volume,[...] created with the column vectors increases compared to the one created with canonical vectors, and I also know that to normalize the vector I can divide it by the norm of vectors of the matrix, so I am asking myself why it works when I use the the nᵗʰ root of the determinant in the place of the norm to get the normalized form of the matrix?
Which number you should divide by depends on what you want, i.e. what exactly it means for a matrix to be "normalized" in your context.
If you want a matrix whose norm is $1$, divide by the matrix by its norm. If you want a matrix whose determinant has magnitude $1$, divide the matrix by the $n$th root of its determinant.