线代试卷-期末.doc

Linear Algebra Final Exam (A)
2004-2005

1. Filling in the blanks (3’×6=18’)
(1) Let be (4×4) 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 (3×3) 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) (2’×6=12’)
(1) If A and B are symmetric (n×n) matrices, then AB is also symmetric. ( )
(2) If A and B are nonsingular (n×n) matrices such that A2=I and B2=I, then, (AB)-1=BA ( )
(3) If u·v =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 (n×n) matrices, then det(A+B)= det(A)+det(B). ( )
(6) If A is an (n×n) 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 ,
(Ⅰ) Find the numbers a and b.
(Ⅱ) Find the eigenvalues of the matrix A.

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