Page 838 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(b) How is the complex number related to the determinant of ?

(c) Show that corresponds to .

(d) Show that the product corresponds to the matrix that is the product of the matrices corresponding
to and .

7.
(a) Prove that if is a complex number other than 1, then

Hint Let be the sum on the left side of the equation and consider the quantity .

(b) Use the result in part (a) to prove that if and , then .

(c) Use the result in part (a) to obtain Lagrange's trigonometric identity

for .
Hint Let
.

Let . Show that the vectors , and form
8. an orthonormal set in . , then the product

Hint Use part (b) of Exercise 7.

Show that if is an unitary matrix and
9.

is also unitary.

Suppose that .
10.

(a) Show that is Hermitian.

(b) Show that is unitarily diagonalizable and has pure imaginary eigenvalues.