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Also, do please let me know if I'm misunderstanding your definition of `B`, because I have a feeling that I might be interpreting it incorrectly.
Nevermind I found it in your comment on the [math.SE post][1] > I can show that expected trace is equal to $(1- \frac{1}{d})^d$ ... So for large $d$, $(1- \frac{1}{d})^d \to \frac{1}{e}$. [1]:...
Also, as another note: Since you are just doing least squares with a square, nonsingular matrix, you could just do x = Inverse[m].b; or more generally (for a non-square matrix): x = PseudoInverse[m].b; to get the least squares...