Ordinary least squares estimates are derived from minimizing the sum of the squared residuals. It can be proven that this minimisation leads to 
. It should be noted that we need to compute an inverse, and that can only be done if the determinant is different from zero. The determinant will be zero if there is a linear dependency between the variables.
It can also be proven that the beta coefficients are distributed according to a Gaussian distribution with variances equal to the diagonal elements of 
where 
is the estimated residual standard error.
estimates using both the
