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Let $X:=\mathbf x$. For each nonrandom vector $a$, the real-valued function $g$ defined on the set of all symmetric positive-definite matrices by the formula $g(A):=a'A^{-1}a$ is convex. So, by Jensen's inequality, $$a'E(XX')^{-1}a=Ea'(XX')^{-1}a=Eg(XX')\ge g(EXX')=a'(EXX')^{-1}a.$$ So, $$E(XX')^{-1}\ge(EXX')^{-1},$$ as desired.