Three factor multiple regression from Snedecor and Cochran (1967), table 13.10.1, page 405.

            Y = estimated plant available phosphorus in the soil (20 C)

            X₁ = inorganic phosphorus

            X₂ = organic phosphorus soluble in K₂CO₃ and hydrolized by hypobromite

            X₃= organic phosphorus soluble in K₂CO₃ and NOT hydrolized by hypobromite

All least squares regression analyses start with the same three matrices. 

            Y´Y =  [ 131299 ]

Create a fully augmented matrix of the form; 

The resulting matrix contains;

Numerically for this problem given previously the matrix is;

The first step (divide row 1 by value_1,1) in the sweepout technique produces,

And after sweeping out the first column (subtracting a multiple of row 1 from all other rows) we have;