Math 110 FINAL ExamProf. BerlekampDec. 10, 1999Use big Blue Books!Problem Parts ? = Points1: Definitions 21 2 = 422: LDU 5 5 = 253: Triangle ABC 2 5 = 104:Pe24 5 = 205: eBt1 10 = 106: Circulant 3 5 = 157: True/False 13 1Prove/Disprove 13 3 = 528: A(x, y)determinant 6Plot 4Shade 3 = 15integers 2Total = 18911. Assume that fields, vector spaces, and matrices have already been defined.Then define each of the following. Be accurate and succinct.A) linearly dependent L) eigenvalueB) span M) eigenvectorC) basis N) generalized eigenvectorD) dimension of vector space O) similar matricesE) rank P) diagonal matrixF) null space Q) diagonalizable matrixG) inverse R) hermitian matrixH) transpose S) unitary matrixI) hermite T) Markov matrixJ) trace U) permutation matrixK) determinant of real matrix2. (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 maindiagonal, and M =1 2 12 6 01 0 5.2(B). Find vectors ~w, ~v, and ~x s.t. L ~w =~b, D~v = ~w, U~x = ~v.Verify that M~x =~b where~b =102.2(C). State whether or not the following equation can be solved by a real vector, ~y,and explain.~yTM~y = −123. 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). ExpressmXi=1e2iin terms of A, ~x, and~b.4(B). Assume that ~x is a differentiable function of time, t, and that ~v =d~xdt.ExpressddtmXi=1e2iin terms of A, ~x,~b and ~v.4(C). Find a sufficient condition which ensures that for all ~v,ddtmXi=1e2i= 0.4(D). Prove this condition is necessary.5. Let B =λ 10 λ, and let t be a scalar. Compute the 2 × 2 matrix eBt=∞Xn=01n!(Bt)n.6. Let C be this real 5 × 5 “circulant” matrix:C =a b c d ee a b c dd e a b cc d e a bb c d e a36(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 = e2πi/5, andverify 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 determinantis the product of the eigenvalues. You may use Perron’s Theorem (which states thatthe largest eigenvalue of a positive matrix is positive). Otherwise avoid unnecessarilyadvanced assumptions.Definitions for [A–E]. A “zero/one” matrix is a matrix each of whose entries iseither a zero or a one. A permutation matrix is a zero/one matrix. There are 16 2 × 2zero/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 sameeigenvalues, 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.4G) Let G be a real symmetric 3× 3 matrix whose entries are all positive, and for whichdet(G) = −1 and Trace(G) = +10If the eigenvalues of G are λ1, λ2, and λ3, and if λ1≥ λ2≥ λ3, then λ3< 0 < λ2< λ1.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 = SΛAS−1and B = SΛBS−1, then AB = BA.K)a bc dandd cb aare similar matrices.M) There exists a 3 × 3 Markov matrix, M, such that M.2.4.2=.1.2.1T) If T is triangular and THT = T TH, then T is diagonal.U) The product of unitary matrices is unitary.8. Consider this real 3 × 3 symmetric matrix: A =1 x yx 1 xy 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