Will $X_{p*n}X_{n*p}'$ be more likely to be invertible when n gets larger? I know $n$ should be at least no less than $p$.
If so, could anyone intuitively or rigidly prove it?
The background of this question is during ratiocination of LSE, say X is fixed and $Y$~$N(X'\beta,\sigma^2)$, to minimize $g(t)=E(Y-X't)^2=\sigma^2+(t-\beta)'XX'(t-\beta)$, we should have more $n$ to get unique $(XX')^-$ so that we can get unique $t^*$ as a minimum point.