Principles of Mathematics in Operations Research

218 Solutions 3.2 V = A) + P\x + e =» E[y) =/3 0 + 0ix. Data: yi= Po + P\x\ 2/2 = Po + P\x 1 Vm = Po + P\Xm & 1 X\ 1 x 2 _ J ...

Solutions 219 SSXX = ]P x^2 - 2mx^2 + mx^2 = y^x; 2 -2 mx , Pi — -mSSxy __ - E x» E 2/i + TO ExiVi which is dictated by the matr ...

(^220) Solutions $• 00 0i 1 = (A<A)-Wy=^ 11 -3 11111 12345 0 = 0o 0i = {A^1 A)^A*y 10 " 11 -3' -3 1 "20" 72 = '0.4' 1.2 = '0o ...

Solutions 221 (b) Let us take the first three columns of A 2 as the basis: B "2 13" 1 32 32 1 , N = "10" 01 _10_ , XB Xi X2 Xi , ...

222 Solutions {AIM -i AT A^1 = 107 90 _iZ 45 47 45 83 90 147 10 2 58 11 J 7 1_ 2 10 15 30 5 4 13 J 3_ 5 30 15 10 x = (AUsr 1 A T ...

Solutions 223 P 3 a:-6|| = error. [2.4201 4.3503 6.2805 [8.2107 "2" 5 6 .8. = 0.4201 -0.6497 0.2805 0.8495 = 0.8695 is the m ...

224 Solutions Problems of Chapter 4 4.1 In order to prove that det A = an An + a^A^ H h ainAin, (property 11) where Aij's are co ...

Solutions 225 Let Cnn — Yz- Thus, det A = an- a 22 • fe • • • (nn = au -Q12Q21 +022011 an 012023031 + 013Q32Q21 ~ Q11023032 ~ Q1 ...

226 Solutions 4.2 Let A = "1 1 2 1 0 1 1 -1 2 -2 -1 1 1 1 2 -1 2 0 3 2 -1 1 -1 1 4 d(s) 2)^5 , fc = 1, Ax =2, m =5. Ai = A - 11 ...

Solutions 227 A 2 = Al = (A--lf = 00^ 00 0 00 0 => dimM(Ai) = 3 - rank(Ai) = 3-1 = 2. A\ = 0 => rfim7V(yl?) = 3 =>• m = ...

228 Solutions where A = ' 0 0 1 20 3 L 100 0 0 1 50 0 3 100 1 25 0 0 1 -1 50 1 100 0 0 and the initial condition is W 0 = [100,6 ...

Solutions 229 '*!(*)* X 2 (t) Yi(t) Mt) .46791 -.46791 -.20890 -.20890 .54010 -.54010 .69374 .69374 .64713 .64713 .33092 -.33092 ...

230 Solutions Problems of Chapter 5 5.1 Proof. Let Q~lAQ = A and Q~l = QT, yTAy Xiyf + • • • + XnVl x = Qy => R(x) = yTy vi + ...

Solutions 231 2 1 0 1 2 1 0 1 1 = 1 = 106 det(^) > 0! The 3x3 minor, itself, is OK as well, iv. All the pivots (without row e ...

232 Solutions Then, we have V^2 /(*A) 32 2 1 and V^2 f(xB) = 52 2 1 Let us check the positive definiteness of V^2 f(xA) using th ...

Solutions 233 Let us check the boundary denned by xj\ /(0, X 2 ) = l-x\ - X 2 + 19 =» ^^^ = a* - 1 = 0 => Z2 = 1. Since —£.? ...

234 Solutions Problems of Chapter 6 6.1 The norm of a matrix A is denned as ||.4|| = -y/largest eigen value of AT A. If Q is ort ...

Solutions 235 Qx -0.6921 -0.5964 -0.3911 0.7218 -0.5718 -0.3750 0 0.5633 -0.7948 0 0 -0.2734 0 0 0 0 0 0 0.1109 -0.0079 -0.0001 ...

236 Solutions A-j = ReQe = A7 = Q7R7, where 166.4231 0 0 0 0 7.7768 0 0 0 -0.0002 1.0218 0.0001 0 0 0.0001 0.2447 0 0 0 0.0321 0 ...

Solutions 237 xn = A(2)-%I = 4 -6 -6 12 1.5 1.0 Ab = bi - bu = xi - xu -0.5 -0.5 f -3 =HIA,II = \H VlO, ||a;/|| = VT=» \\xi\\ Th ...