Answer: a) Let Q be an orthogonal matrix and let λ be an eigenvalue of Q. Then there exists a non-zero vector v such that Qv = λv. Taking the conjugate transpose of both sides, we have:
(Qv)^T = (λv)^T
v^TQ^T = λv^T
Since Q is orthogonal, we have Q^TQ = I, so Q^T = Q^(-1). Substituting this into the above equation, we get:
v^TQ^(-1)Q = λv^T
v^T = λv^T
Taking the norm of both sides, we have:
|v|^2 = |λ|^2|v|^2
Since v is non-zero, we can cancel the |v|^2 term and we get:
|λ|^2 = 1
Taking the square root of both sides, we get |λ| = 1.
b) Let Q1 and Q2 be orthogonal matrices. Then we have:
(Q1Q2)^T(Q1Q2) = Q2^TQ1^TQ1Q2 = Q2^TQ2 = I
where we have used the fact that Q1^TQ1 = I and Q2^TQ2 = I since Q1 and Q2 are orthogonal matrices. Therefore, Q1Q2 is an orthogonal matrix.