In the notation of Problem 9, demonstrate that two positive definite matrices A = (aij ) and B = (bij ) satisfy AB if and only they satisfy B−1  A−1. If AB, then prove that det A ≥ det B, tr A ≥ tr B, and aii ≥ bii for all i. (Hints: AB is equivalent to xt Ax ≥ xtBx for all vectors x. Thus, AB if and only if I A−1/2BA−1/2 if and only if all eigenvalues of A−1/2BA−1/2 are ≤ 1.)