线代试卷 期末.doc

Linear Algebra Final Exam (A)2004-20051. Filling in the blanks (36=18)(1) Let be (44) matrices, and det(A)=4, det(B)=1, then det(A+B)= .(2) Let , then .(3) Let is a n dimension vector, and , then the matrix AB= .(4) Let A be a (33) matrix, and 1,2,3 are the eigenvalues of A. Then the eigenvalues of A* are .(5) Let be a linearly dependent set of vectors, where . Then the number k is .(6) Let . Then the cross product = .2. Determining the following statement whether it is true(T) or false(F) (26=12)(1) If A and B are symmetric (nn) matrices, then AB is also symmetric. ( )(2) If A and B are nonsingular (nn) matrices such that A2=I and B2=I, then, (AB)-1=BA ( ) (3) If uv =0, then either u =0 or v =0 . ( )(4) If A is nonsingular with A-1=AT, then det(A)=1 ( )(5) If A and B are diagonal (nn) matrices, then det(A+B)= det(A)+det(B). ( )(6) If A is an (nn) matrix and c is a scalar, then det(cA)=cdet(A).( )3. (15)Calculate the determinant of the matrix .4. (15)Consider the system of equations , determine conditions on k that are necessary and sufficient for the system to be has only solution, infinite solutions, and no solution, and express the solutions by vectors.5. (10)Let be a set of nonzero vectors in Rm such that when . Show that the set is linearly independent.6. (15)Let and . Find B.7. (15)Let is an eigenvector of ,