大二上期末复习-职协cic线性代数exercise

1、1The First Examination1. Consider the system of equations For what values of a does the system have infinitely many solutions? No solutions? A unique solution in which x2=0?Solution: For augment matrix, set row operations(1)a2+a-2=0 and a2-a-6=0,that is, a=-2, infinitely many solutions.(2) a2+a-2=0

2、but a2-a-60,that is, a=1, no solution.(3) a2+a-2 0,that is a -2 , a 1, unique solution.and a -2 , a=3, unique solution(in this situation x2=0).22. Let Determine conditions on b1,b2,and b3 that are necessary and sufficient for the system if equations AX=b to be consistent.For each of the following ch

3、oices of b, either show that the system AX=b is inconsistent or exhibit the solution.33. Let anda)Simultaneously solve each of the systems Ax=bi,i=1,2,3, whereb)Let B=b1,b2,b3. Use the results of part(a) to exhibit a (33) matrix C such that AC=B. Solution:44. Let Find a (32) matrix B such that AB=C.

4、 Solution:55. Let A be the nonsingular (55) matrix, and let . For a given vector b, suppose that 1,3,5,7,9T is the solution to Bx=b. What is the solution of Ax=b? Solution:66.Let a) Solve the vector equation , where b) Show that the set of vectors v1,v2,v3 is linearly dependent by exhibiting a nontr

5、ivial solution to the vector equation Solution:77. Let And define a function T:By T(X)=AX for each in R3(a) Find a vector X in R3 such that T(X)=b, where (b) If 0 is the zero vector of R3, then clearly T(0)=0. describe all vectors X in R3 such that T(X)=0.88. Let FindSuch that 99.Find A-1 for each o

6、f the following matrices A (1)(2)1010.For what values of is the matrixsingular? Find A-1 if A is nonsingular.1111.Find A if A is (22) matrix and 1213. Let Calculate A99 and A1001318.Let a) Without calculate A or B, find (ATB)-1b) Without calculate A or B, find(A-1B-1)-1A-1B-1 14The First Examination

7、1、A is nn matrix，A=A1 A2 A3，Ai is the ith column of A, so Solution:152、A is 33 matrix,and ,so(a)4, (b)-4, (c)16, (d)-163、A is 33 matrix ，A* is the adjoint matrix of A, and ，please calculate the value of Solution:Solution:164、calculate the determinant ofExpanded on the last row17185、if，findSolution:1

8、96、Solution:207、Solution:218、Solution:229、If A and B are nn matrices ，then（ ）A、|-A-1|=- |A-1|-1 B、(AB)K=AKBKC、|A|B|=|B|A| D、|A*|=|A|n-110. If A,B,C, are nn matrices, such that ABC=I，then（ ）A、 ACB=I B、 CAB=I C、 BAC=I D、BCA=I2311.If A and B, are nn matrices, then（ ）If A or B is invertible ，so is AB If

9、 A or B is not invertible ，then AB is not invertible(c) If A and B are invertible ，then A+B is invertible (d) If A and B are not invertible, then A+B is not invertible 2412.Suppose,Then （ ） A、AP1P2=B B、 AP2P1=B C、 P1P2A=B D、 P2P1A=B 2513、If Where ai0,i=1,2,n ,find A-1Solution: we apply the formula 2

10、6If ，where P is a 33 invertible matrix，then B2004-2A2=Solution: because2715.If then k0, k316. supposeIf k1/4，then 123 are independent Solution: becauseSolution: because2817. Suppose the 3 dimension vectors aredetermine the value of ,which makes the following statement true. （1） is the only combinati

11、on of 1 2 329(2） could be expressed the linear combination of 1 2 3 ,and the combination is not only.（3） could not be expressed the linear combination of 1 2 3When =0, Because When =-3, Because 3018. Suppose A=(aij)is 33 matrix,satisfy （1） aij= Aij(i,j=1,2,3),where Aijis the factor of aij（2）a110Plea

12、se calculate |A|Solution : because aij= Aij , so AT=A*And AA*=|A|3ISo |A| |A*|= |A|3 |A|2= |A|3 |A|2(|A|-1)=0 |A|=0 or |A|=1But a110, so |A|=13119.Suppose A is n m matrix, B is m n matrix，and nm，I is n n identity matrix,satisfy AB=I .Show that ：the column vectors of matrix B is independent.Proof :Su

13、ppose So 3220.Suppose the set 1,2 , ,m-1 (m3) is linear dependent，but the set 2 , ,m is linear independent，discuss： (1) Is 1could be written as the linear combination of 2 , ,m-1？ (2)Is m 1could be written as the linear combination of 1,2 , ,m-1？33The second examination(1) if is a eigenvector of A，t

14、hen we deduce a eigenvector of matrix P-1AP is (2) If A is ( ) zero matrix, then its all eigenvalues are 34(3) If A is ( ) matrix, are two eigenvalues of A, and are the eigenvectors corresponding to ,respectively. Then ( )a. When , must be proportional b. When , must be not proportionalc. When , mus

15、t be proportionald.when , must be not proportional 35(4) Determine the eigenvalues of A, where A is 36(5) If A is (44) matrix, and A satisfy |3I+A|=0 ，AAT=2I ，|A|0. Determine a eigenvalue of A*, where A* is the adjoint matrix. 37(6) If A is (33) matrix, and A satisfy , where , . Please determine mat

16、rix A. 38(7) Suppose (33) matrices , . Determine whether matrix A is similar to matrix B. if yes, calculate the nonsingular matrix M, such that 39(8) Suppose matrix . Determine the value of parameter k, such that P-1AP is diagonal matrix. Then calculate the nonsingular matrix P and the diagonal matr

17、ix 40(9) Suppose A is (nn) orthogonal matrix, and |A|=-1. Show that： is an eigenvalue of A. 41(10) Suppose A and B are (nn) matrices，matrix A is similar to matrix B, then ( )a. b. A, B have the same eigenvalues and the same eigenvectorsc. A,B are similar to the same diagonal matrix d. For arbitrary

18、constant t, matrix A-tI is similar to matrix B-tI, where I is the identity matrix. 42(11) If matrix , and |A|=-1. Suppose is the eigenvalue of adjoint matrix A*, is the eigenvector of . Determine parameters a,b,c, and 43(12) Suppose A and B are (44) matrices, and A is similar to B, the eigenvalues of A are . Determin