複選題

二、 Which of the following statements about the multiplicative inverse of a matrix is/are true?
(A) A matrix A is called invertible if there exists a matrix B such that AB is an identity
matrix.
(B) If a matrix is both diagonalizable and invertible, then so is its multiplicative inverse.
(C) Suppose that matrices A of size n x n and D of size m x m are invertible and that
matrix C is of size m X fi； then the following identity is true

(D) Methods for finding the multiplicative inverse of a matrix include LU factorization,
Gaussian elimination, eigen-decomposition, and Gram-Schmidt process.
(E) None of the above are true.

48.假設全世界只接受美元為國際支付的工具, 美金是唯一的外匯, 請問 下列何者不構成對外匯的供給? (A) 四毛在美國攻讀博士, 期間也投資台灣 股市, 他以 1 千美元投資於心目中有潛力的台灣上市公司。 (B)五毛為美 商麥當勞的經理, 日前收到美國匯來的 500 萬元營運資金。 (C)三毛長期 在美國工作, 今年農曆過年回台灣團聚, 攜回 7 千美金發壓歲錢。(D) 大毛 出售自己撰寫的小說 200 本到香港, 總共收得美金 2000 美元。 (E) 二毛 將選舉結餘經費 1 億新台幣換成美元匯入小兒子在美國舊金山的帳戶。

49.假設美國投資人看好我國股市前景,對台灣投資資金 5000 萬美元, 請問在台幣/美元的外匯市場上會發生何種變動? (A) 外匯的供給曲線右移。  (B)匯率不變。 (C)台幣相對於美元會貶值。(D) 美元相對於台幣升值。 (E) 外匯的需求曲線左移。

50.財政部開放外國人投資台灣的股市, 將外國人投資每家上市公司的上 限由 25% 提高到 50 %, 此舉造成外資大量湧入台灣股市； 在其它條件不變 下(中央銀行亦未干預), (A) 市場利率將大跌。 (B)股市泡沫將造成房地產 大跌價。 (C)新台幣將因此貶值。(D) 市場利率將大漲。(E) 新台幣將因 此升值。  

一、 Which of the following sets is basis (are bases) for ? (A) .(B) (C) (D) (E)None of the above are true.

複選題三、Which of the following properties of eigenvalue is/are true? (A) A scalar λ is an eigenvalue of matrix A if and only if λ is an eigenvalue of .(B) A matrix is positive semi-definite if and only if all of its eigenvalues are non-negative.(C) Every eigenvalue of a matrix A is also an eigenvalue of A2.(D) If matrices A and B are similar, then they have the same eigenvalues.(E) None of the above are true.

四 、 Which of the following matrices is/are diagonalizable? (A) (B) (C) (D) (E) None of the above are true. 

複選題五、Which of the following statements about matrix factorization is/are true? (A) If a matrix A is positive definite, then A has "an" LU factorization, A = LU, where the diagonal entries of U are positive.(B) Suppose that a matrix A = QR, where Q is an m x n matrix and R is an n X n matrix. If the columns of A are linearly independent, then R must be invertible.(C) Any factorization of a matrix A = , with matrices U, V square and positive diagonal entries in the matrix D, is called a singular value decomposition of A.(D) An n x n matrix A is positive definite if and only if A has "a" Cholesky factorization A = for some invertible upper triangular matrix R whose diagonal entries are all positive.(E) None of the above are true.

複選題六、 The Householder matrix H = is the orthogonalprojection matrix onto some nonzero vector , is a reflection matrix. Which of thefollowing statements is/are true?(A) Consider the 2-dimensional space, ie. n = 2. For vectors u, d, p,q   shown in thefigure below, we have H d = p.(B) H is a symmetric and orthogonal matrix.(C) Both linear systems A z = o and HAz = Ho are equivalent.(D) Let a = ≠0. Then llull2 = -2 llαll u1and Hα = (E) None of the above are true.

七、 Given029which of the following statements is/are true?(A) (B) (C) (D) det(F) = 20.(E) None of the above are true.

八、 Consider a 3 x 3 nonzero matrix A, and let W = . The leastsquares solutions xLs to Ax = b are illustrated in the following figure，where projwe b is theorthogonal projection of vector b onto vector space W. A minimun length least squares, solution EMurs is the one among the least squares solutions that has a minimum norm.Which of the following stateme ents is/are true?(A)(B) (C) (D) The minimum length least squares solution EMLLS to Ax = b is unique, and =projvxLs, where V = .(E) None of the above are true.