MAT 215 Final, Spring 2001

MAT 215 — Linear Algebra I Spring 2001 — Final Exam Duration: 2 hours iy —| (2) I)Let 4=]1 0 a jjand b=)1 pee summa) 1) Give conditions ona so that Ax = has: a) A unique solution b) More than one solution c) No solution (15 points) 2 A @ 2)Let A=]0 -1 2]. ie 5 a) Compute 4’, 4? and show that 47> =94-8/. b) Deduce that 4 is invertible and express 4”! in terms of 42 and c) Find A”. (15 points) (! -1 : wl +1) 3} Let 4 ==! ibend B=! 0 i =H l, G 4 lo KY @) Prove that 4 is row equivalent to B b) Find element (10 points) 4) Use Gaussian elimination to compute the determinant of ft -1 2 1 0) l a ay || ] { { — = Nm | ! XN (10 points) 5) Let HY = {(es)e R [ > of be a subset of R?. a) ‘Prove that Ff’ is closed under scalar multiplication y matrices £),°E,) E, so that B= EXE,E, A. b) Find a specific example to show that His neta sub sPacp a V, (10 points) auf 3 SF 0 3 2 a) Pind the rank of A and deduce that the subset S={v, =(L-111),v, =(2, 11,3). =(4,2,-2,1,v, =(3,-3,3, 2)} is linearly dependant. b) Find a basis B for lin(S) such that Bisa subset of S. (10 points) olen 7)Let 4=/2 2 2], 0 0 0 a) Find the eigenvalues and eigenvectors of A. b) Is A diagonalizable? Why? c) Find an invertible matrix P such that P“\AP = matrix. d) Calculate der(4'°) and r(A'opf D where D isa diagonal (20 points) S) Let 4 be an invertible + ax matrix and 2 be an eigenvalue of 4. show that 27* 5 =J an eigenvalue of A7'. is (19 points)