线代英文试卷-2004-2005 Linear Algebra Final Exam A.doc

1 Linear Algebra Final Exam (A) 2004-2005 1. Filling in the blanks (3’× 6=18 ’) (1) Let],,,[ and ],,,[ 432432??????????BA be (4× 4) matrices, and det(A)=4, det(B)=1, then det(A+B)= . (2) Let???????????523 012 101A , then ???1)(AI . (3) Let???????????????????2/1 0 0 2/1?? isan dimension vector, TIA ???? and TIB ?? 2??, then the matrix AB =. (4) Let A bea (3× 3) matrix, and 1,2,3 are the eigenvalues ofA. Then the eigenvalues ofA * are. (5) Let},,{ 321????S bea linearly dependent set of vectors, where ????????????????????????????????????3 2 1,1 3 2,2 4 0 321kkk ???. Then the number k is. (6) Let??????????????????????0 1 4,0 3 1??. Then the cross product ???=. 2. Determining the following statement whether it is true(T) or false(F) (2’× 6=12 ’) 2 (1) IfA and B are symmetric (n× n) matrices, then AB is also symmetric. () (2) IfA and B are nonsingular (n× n) matrices such that A 2 =I and B 2 =I, then, (AB) -1 =BA() (3) Ifu· v =0, then either u =0 orv =0 .() (4) IfA is nonsingular with A -1 =A T, then det(A)=1 () (5) IfA and B are diagonal (n× n) matrices, then det(A+B)= det(A)+det(B). () (6) IfA is an (n× n) matrix and c isa scalar, then det(cA)=cdet(A).( ) 3. (15 ’) Calculate the determinant of the matrix ??????????3214 2143 1432 4321 . 4. (15 ’) Consider the system of equations ????????????