Start with an orthonormal matrix and multiply it by its transpose. When the ith row of M is multiplied by the jth row of MT, that's the same as taking the dot product of rows i and j. This is 1 when i = j, and 0 otherwise. Guess what, the product is the identity matrix. The transpose is the inverse.
If M is orthogonal, and the ith vector has length s, multiply M by its transpose and the element Pi,i is s2. We want it to be one. Divide the ith column of the transpose, or the ith row before you perform the transpose, by s2, and this little problem goes away. Do this for every row and find the inverse of the orthogonal matrix. The inverse is the transpose, but each row has been divided by its length squared.
Let M be orthonormal once again. Since M*MT is the identity matrix, as shown above, the determinant squared = 1. Therefore the determinant of an orthonormal matrix is ±1. The sign depends on the order of the rows. For instance, write down the x y and z unit vectors, in that order, and obtain a 3×3 matrix with ones down the main diagonal. This is an orthonormal matrix with determinant 1. Swap the last two rows and find an orthonormal matrix with determinant -1.
By scaling rows, the determinant of an orthogonal matrix is ± the product of the lengths of its rows.
All this can be generalized to complex numbers. Multiply M by its tranjugate M*, rather than its transpose MT. This produces a series of dot products as before, and the result is the identity matrix. Scale the rows of M, and the inverse of an orthogonal matrix is its tranjugate, with its columns scaled, as described above.