[donuteaters]

You calculate the factorization (QR, LU, or similar), but you never calculate the inverse of the factorization:

    A x = b
    A => QR  # QR decomposition
    Q R x = b
    R x = Q' b  # Q is inverted by transposing
    x = backsubstitute(R, Q' b)

Q was the transpose of it's inverse, and so you don't get any rounding error when you transpose it. R didn't need to be inverted because it was in a form where back substitution lets you apply the inverse of R to be without needing the inverse of R.

[pramodliv1]

MATLAB's linsolve uses your strategy! (http://in.mathworks.com/help/matlab/ref/linsolve.html)

[ryanmonroe]

Linear models in R are also solved by QR factorization.