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Linear Algebra Final Test (A)2005-20061. Filling in the blanks (36=18)(1) Let be (22) matrices, and det(A)=3, det(B)=-2, then det(A+B)= .(2) Let , then .(3) Let is a 4 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 I+A* are .(5) Let be a linearly dependent set of vectors, where . Then the scalar 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 is an (mn) matrix such that AX=0 for every X in ,then A is the (mn) zero matrix. ( )(2) If A and B are nonsingular (nn) matrices then AB 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 eigenvalues. ( )(6) If A is an (nn) matrix such that det(A)=1,then AdjAdj(A)=A. ( )3. (15) Evaluate the determinant of the matrix .4. (15)Consider the system of equations , determine conditions 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.5. (10)Let U and V be nonzero vectors such that . show that U-V and U+V are orthogonal.6. (15)Let and . Find B.7. (15)Let is an eigenvector of ,