試題詳解

9、Which of the following statements is/are true?
(A) Let x and y be two non-zero vectors in Rᵀ. The matrix xyᵀ is diagonalizable if
yᵀx ≠ 0.
(B) If A ∈ Rⁿˣⁿ is symmetric and positive definite, then A + A⁻¹ - 2Iₙ is positive semi- definite.
(C) For u, v ∈ Rⁿ such that uᵀv ≠ 0, Rⁿ = (span{u})^⊥⊕span{u}, where ⊕ is the
operation of direct sum of two vector spaces.
(D) For the real matrix Q = Iₙ - uuᵀ, where u ∈ Rⁿ, we have det(Q) + rank(Q) = n.

(E) None of the above is true.