江西财经大学
2009－2010学年第一学期期末考试试卷
试卷代码：12063A 授课课时：48
课程名称：Linear Algebra 适用对象：2008级国际学院
1. Filling in the Blanks (3’×6=18’)
?234?? , then det(adj(A))= . 012(1) If A??????003??
?0?1(2) If A???0??0?0?0(3)If A???b??a100000ab0011ba000?0??, then the inverse A?1 = 1??2?a?b??, then det(A)= 0??0?(4) Let A be (4×4) matrix, and -1,2,4,6 are the eigenvalues of A. Then the eigenvalues of A-1 are . ?10??9?????(5) Let ????12?,????7?. Then the tripe products ??(???)= . ???1???3???112?(6) If the rank of A??213? r(A)=2, then parameter a= .
????1a?1??2. There are four choices in each question, but only one is correct. You should choose the correct one into the blank. (3’×6=18’)
(1) Let A and B be (3×3) inverse matrices, then ( ) is not always correct. (A) (AB)T?BTAT (B) (AB)?1?B?1A?1 (C) adj(AB)?adj(B)adj(A) (D) (AB)2?B2A2
(2) Let A and B be (n×n) matrices, and AB?0,B?0, then .
(A)B?0 (B) adj(B)?0 (C) AT?0 (D) (A?B)2?A2?B2
(3) If ?,? are n dimension column vectors, and ?,? are orthogonal, then ( ) is not always correct.
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(A)(???)?0 (B)?T??0 (C) ?,? are linear independent. (D)???0 (4) Let A be (m?n) matrix，and m?n, then the statement ( ) is always true. (A) AX?0 has infinitely many solutions. (B) AX?b has infinitely many solutions. (C) AX?0 has no solutions. (D)AX?b has no solutions.
(5) If n?n matrices A and B are similar, then the statement ( ) is always true. (A) A,B have the same eigenvalues and eigenvectors. (B) A,B only have the same eigenvectors. (C) The rank of matrices A,B, such that r(A)=r(B)
(D) The column vectors of A and B are all linear independent.
(6) If A is an (3×3) orthogonal matrix, A?[A1,A2,A3], Aiis the column vectors of A, then the statement ( ) is not always true. (A) {A1,A2,A3} is an orthogonal set. (B) {A1,A2,A3} is a linear independent set. (C) {A1,A2,A3} is a basis of R3. (D) A1A2A3?0
?x1?2x2?x3??1?ax2?x3?13. (12’) If the system of linear equations is ?, then what value of a will ?x?3x?(a?1)x?023?1make the system has only solution, infinitely many solutions, no solutions, and when the system has infinitely many solutions, find its all solutions.
4. (12’) Let A?(aij)3?3 matrix such that det(A)=3, and let Aij denote the ijth cofactor of A. If
?A31B???A32??A33A21A22A23A11?A12??, then calculate AB. A13??5. (10’) Suppose A is a (2×2) matrix such that A2?3A?I?0, and Au???,whereu???.
?1??3?Find A2uandA3u.
?2??1??2?20??. 6. (15’) Find the all eigenvalues and all eigenvectors of matrix A, where A???21?2????0?20??第 2 页 共 3 页
7. (10’) There is a linear independent vectors set {?1,?2,?3}, where
?0??0??1??1??1??1??1???,?2???,?3???. Please generate an orthogonal set{u1,u2,u3}.
?2??3??1???????11?????0?8. (5’) Suppose A is (n×n) orthogonal matrix, and |A|=-1. Show that：???1 is an eigenvalue of
A.
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