Math 110 FINAL Exam Prof Berlekamp Dec 10 1999 Use big Blue Books Problem Parts 1 Definitions 2 LDU 3 P Triangle ABC 4 e2 Bt 5 e 6 Circulant 7 True False Prove Disprove 8 A x y determinant Plot Shade integers 21 2 5 5 2 5 4 5 1 10 3 5 13 1 13 3 6 4 3 2 Points 42 25 10 20 10 15 52 15 Total 189 1 1 Assume that fields vector spaces and matrices have already been defined Then define each of the following Be accurate and succinct A linearly dependent B span C basis D dimension of vector space E rank F null space G inverse H transpose I hermite J trace K determinant of real matrix L eigenvalue M eigenvector N generalized eigenvector O similar matrices P diagonal matrix Q diagonalizable matrix R hermitian matrix S unitary matrix T Markov matrix U permutation matrix 2 A Factor the following real 3 3 matrix into M LDU where L is lower triangular D is diagonal U is upper triangular and both L and U have all ones on the main 1 2 1 diagonal and M 2 6 0 1 0 5 2 B Find vectors w v and x s t b D v w U x v Lw 1 Verify that M x b where b 0 2 2 C State whether or not the following equation can be solved by a real vector y and explain y T M y 1 2 3 Consider the triangle ABC formed by these points in R3 A 2 4 2 B 4 6 4 C 6 6 4 3 A Find the angles of this triangle 3 B Find the area of this triangle 4 Let A be an m n matrix b an m 1 vector and x an n 1 vector Define e A x b A b x and e are all real 4 A Express m X e2i in terms of A x and b i 1 d x 4 B Assume that x is a differentiable function of time t and that v dt m d X 2 Express e in terms of A x b and v dt i 1 i 4 C Find a sufficient condition which ensures that for all v d dt m X i 1 e2i 0 4 D Prove this condition is necessary 5 Let B X 1 1 and let t be a scalar Compute the 2 2 matrix eBt Bt n 0 n n 0 6 Let C be this real 5 5 circulant matrix a b e a C d e c d b c 3 c b a e d d c b a e e d c b a 6 A Exhibit a permutation matrix P and a real polynomial f x s t C f P 6 B Write down the 5 5 Fourier matrix F explicitly in terms of w e2 i 5 and verify that the columns of F are the eigenvectors of P 6 C Express the eigenvalues of C in terms of f x and w 7 13 parts For each of the following assertions state whether it is True or False Then Prove or Disprove the assertion You may use the fact that the determinant is the product of the eigenvalues You may use Perron s Theorem which states that the largest eigenvalue of a positive matrix is positive Otherwise avoid unnecessarily advanced assumptions Definitions for A E A zero one matrix is a matrix each of whose entries is either a zero or a one A permutation matrix is a zero one matrix There are 16 2 2 zero one matrices A A real 2 2 zero one matrix must have real eigenvalues B If the eigenvalues of a real 2 2 zero one matrix are real they must be integers C If G and H are nonsingular real 2 2 zero one matrices which have the same eigenvalues then they must be similar Definitions for D E Let A be a real 3 3 zero one matrix and let B be the same binary matrix Real 1 1 2 0 binary 1 1 0 D If A is invertible then B is invertible E If B is invertible then A is invertible F If CD DC then C or D must be singular 4 G Let G be a real symmetric 3 3 matrix whose entries are all positive and for which det G 1 and Trace G 10 If the eigenvalues of G are 1 2 and 3 and if 1 2 3 then 3 0 2 1 H The product of Hermitian matrices is Hermitian J If A and B are diagonalizable and if they share the same eigenvector matrix S such that A S A S 1 and B S B S 1 then AB BA K a b c d and d c b a are similar matrices 2 1 M There exists a 3 3 Markov matrix M such that M 4 2 2 1 T If T is triangular and T H T T T H then T is diagonal U The product of unitary matrices is unitary 1 x y 8 Consider this real 3 3 symmetric matrix A x 1 x y x 1 A Express determinant A as a polynomial in x and y B Plot all points in the x y plane where determinant A 0 C On another plot of the x y plane shade the region s where A is positive definite D List as many pairs of integers x y for which A is positive definite as you can But do not list more than ten such pairs 5