388 CHAPTER 13. MATRICES AND THE INNER PRODUCT

where A11,A22 are square matrices. Show that det(A) ≤ det(A11)det(A22). Hint:Use the above problem to factor A getting

A =

(B∗11 0∗

B∗12 B∗22

)(B11 B12

0 B22

)

Next argue that A11 =B∗11B11,A22 =B∗12B12+B∗22B22. Use the Cauchy Binet theoremto argue that det(A22) = det(B∗12B12 +B∗22B22)≥ det(B∗22B22) . Then explain why

det(A) = det(B∗11)det(B∗22)det(B11)det(B22)

= det(B∗11B11)det(B∗22B22)

34. ↑ Prove the inequality of Hadamard. If A is a Hermitian matrix which is nonnegative(all eigenvalues are nonnegative), then det(A)≤∏i Aii.