(Schur's Theorem) If $A$ is positive, then $$ex per(A)geq det A. eex$$
Solution. By Exercise I.2.2, $A=T^*T$ for some upper triangular $T$ with non-negative diagonals. Thus $$eex ea det A&=det T^*cdot det T\ &=per T^*cdot per T\ &=per(T^*I)cdot per(Icdot T)\ &leq sqrt{per(T^*T)cdot per (I^*I)}cdot sqrt{per(II^*)cdot per (T^*T)}\ &=per(T^*T)\ &=per(A). eea eeex$$