試題詳解

34. Let R_θ denote the rotation matrix through an angle 0＜θ＜counterclockwise about the origin in R². Let e₁ = [1,0], e₂ = [0,1]ᵀ be the standard basis of R². Suppose A is a 2x2 matrix satisfying Ae₁ = R_θe₁ and R_θAe₂ = e₂. The followings are 5 statements about A that may or may not be true.
(1) A²e₁ = R₂θe₁
(2) det( A ) = 1
(3) trace( A ) = 2cos θ
(4) A is a positive definite matrix
(5) AR_θ = R_θA
Which of the following combination of statements about A are all true?
(A) (1)(2)
(B) (1)(3)
(C) (2)(3)
(D) (3)(4)

(E) (4)(5)