Page 835 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(b) What can you say about the inverse of a matrix that is both Hermitian and unitary?

Prove: If is an matrix with complex entries, then .
15.

Hint First show that the signed elementary products from are the conjugates of the signed elementary products from .

16. matrix with complex entries, then .
(a) Use the result of Exercise 15 to prove that if is an

(b) Prove: If is Hermitian, then is real.

(c) Prove: If is unitary, then .

Prove that the entries on the main diagonal of a Hermitian matrix are real numbers.
17.

Let
18.

be matrices with complex entries. Show that
(a)
(b)
(c)
(d)

Prove: If is invertible, then so is , in which case .
19.

Show that if is a unitary matrix, then is also unitary.
20.

Prove that an matrix with complex entries is unitary if and only if its rows form an orthonormal set in with the
21. Euclidean inner product.