Problem 1. (10 %) Given a matrix

where a is a real number. Find the rank ofand give the corresponding range of a.

5.試著說明美國2017年國家安全戰略報告為何將中國大陸與俄羅斯謝為最重要的對手？

6.從我國的角度來看，釣魚臺列嶼在哪裡？

7.美國為了給予南海主權之爭的國家戰略保證所採取的「航行自由行動」，可能產生哪些影響？其中，我國可以做些什麼？

8.「中」印邊界衝突擴大可能對我國帶來什麼影響？

(i) (10 %) Find the Eigenvalues and Eigenvectors for .

(ii) (5 %) Show that there exists a matrix, which is similar to the matrixin the Jordon form, i.e., 

(i) (15 %) Letwith r singular valuesUr= [u1,⋯,ur]consisting of the associated r left singular vectors and Vr= [v1,⋯,vr]consisting of the associated r right singular vectors. (1) Find the range space (i.e., column space) of A, denoted as R(A), and the row space of A (i.e., R(AT)) where AT denotes the transpose of A ；(2) find the projection matrix PA such thatand；(3) find the eignevalues and eigenvectors of AAT.

(ii) (10 %) Suppose that A,and Tr(A) denotes the trace of A (i.e., the sum of all the diagonal elements of A).(1)Is Tr(AX)≥0 true? (2)Is it true that if Tr(AX) = 0, then(zero matrix)?Prove or disprove your answer.

1.國防政策並非局限於____________，任何與維護國防安全的事物都屬於國防政策範疇。

2.非傳統安全威脅與日俱增，國軍秉持________________________原則，守護家園及人民福祉。