MAT 215 Final, Fall 1998

MAT 215 (Linear Algebra, Final Exam, Fall98-99, Duration: 2 hours 1. (15%) For which values of a will the following system have no solutions? Exactly one 5 ( \ ; ye wk} Yes } ° ‘\ solution? Infinitely many solutions? x+2y- 3x-— yr 4x+yt(a’?-14)z=a+2 (15%) Let 4 and B be nxn matrices, and m be any positive integer. Use determinanis to answer the following questions: re a) If AB is invertible, then 4 and B are both invertible. (b) If A” is invertible, then A is invertible. (c) If A” =0, then A cannot be invertible. (d) Ifn is odd and 4B + BA =0, then J or Bis not invertible \(2) Give an example ofa/?x 2 nonzero mawix C whose square CHO. (15%) (a) Determine whether or not the three vectors w=(2,-1, 6), v=(1,-1,1), and aN Lig Br, Sood © i a) (Bw = (0, 2,3) form a basis for the oe or space RY <= See ayy? (o 1) (0 0 1 Saag hes (ye) Prove that the set Ss; = 0 0 {0 o} ti is ee Seale set for the vector space M... & che wily Mans 7. | hl S| é “ost ; ate CF +o beh (6) (c) Extend S, to a spanning set S, of Ms, .< be = (BX) Is S, a basis for .Wf,,? Why or shy not? ~ i= 0 QO - 1| 4. (20%) Consider the matrix 4=)0 1 0| 2| g v ( oe (a) Find the eigenvalues and eigenvectors of 4. (b) Find an invertible matrix P such that PAP is diagonal. (c) Use (a) and (b) to evaluate 4''.