1、1Linear Algebra Final Test (A)2005-20061. Filling in the blanks (36=18)(1) Let be (22) matrices, and det(A)=3, 2,and3,2BAdet(B)=-2, then det(A+B)= .(2) Let , then .2031A1)(AI(3) Let is a 4 dimension vector, and 112 TIA2, then the matrix AB= .TIB(4) Let A be a (33) matrix, and 1,2,3 are the eigenvalu
2、es of A. Then the eigenvalues of I+A* are .(5) Let be a linearly dependent set of vectors, where ,321S. Then the scalar k is .k1,3,12(6) Let . Then the cross product = .1,30 2. Determining the following statement whether it is true(T) or false(F) (26=12)2(1) If A is an (mn) matrix such that AX=0 for
3、 every X in ,then A isnRthe (mn) zero matrix. ( )(2) If A and B are nonsingular (nn) matrices then A B is also nonsingular. ( ) (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 S is (nn) and nonsingular, then A and have the same S1eigenvalues.
4、( )(6) If A is an (nn) matrix such that det(A)=1,then AdjAdj(A)=A. ( )3. (15) Evaluate the determinant of the matrix . bababb4. (15)Consider the system of equations , determine 15422321321xxconditions on that are necessary and sufficient for the system to be has only solution, infinite solutions, and no solution, and express the solutions by vectors.35. (10)Let U and V be nonzero vectors such that . show that U-V and VUU+V are orthogonal.6. (15)Let and . Find B.1023A IBAA3117. (15)Let is an eigenvector of , 1 1243baA() Find the numbers a and b.() Find the eigenvalues of the matrix A.
