On building matrices in the theory of least squares method

We construct samples of real matrices with number of rows greater than the number of columns and satisfying four requirements: the squares of the rows equal one, the squares of the columns equal each other, the columns are pairwise orthogonal, the sum of the components of each column is zero, except for two cases. In the first case, the number of rows is odd and the number of columns is one. In the second case, the number of rows is odd and the number of columns is two less than the number of rows. It is proved that in these cases, there are no matrices satisfying the four specified requirements. The place of matrices satisfying the four specified requirements is shown in the theory of errors in measuring systems such as GPS.