palm_ch6 - Solutions to Problems in Chapter Six Test Your...

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Solutions to Problems in Chapter Six Test Your Understanding Problems T6.1-1 x =2 , y = - 3, z =5 T6.1-2 Multiplying the first equation by - 3 shows that it has the same left side as the second equation, but a different right side. Thus the equations are represented by parallel lines, and no solution exists. T6.1-3 Substitute the second equation into the first to obtain 4 x - b (0 . 5 x )=0or(8 - b ) x = 0. If b 6 = 8, the solution is x = y =0 . I f b = 8 the two equations are identical, and all we can say is that x y . T6.2-1 Solving by hand: A = " 3 - 4 6 - 10 # b = " 5 2 # Thus x = A - 1 x = 1 6 " 10 - 4 6 - 3 #" 5 2 # = " 7 4 # The Matlab session is ± A = [3,-4;6,-10]; ± b = [5;2]; ± x = inv(A)*b x= 7.0000 4.0000 T6.2-2 Solving by hand: A = " 3 - 4 6 - 8 # b = " 5 2 # But | A | =3( - 8) - 6( - 4) = 0, so the inverse of A does not exist. Thus there is no unique solution. (These two equations are represented by parallel lines.) The Matlab session is ± A = [3,-4;6,-8]; ± b = [5;2]; ± x = inv(A)*b Warning: Matrix is singular to working precision. 6-1
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x= Inf Inf Thus no solution exists. T6.3-1 Cramer’s determinants are D = ± ± ± ± ± 4 - b - 36 ± ± ± ± ± D 1 = " 5 - b ± ± ± ± ± D 2 = " 45 - 33 ± ± ± ± ± These determinants evaluate to D =24 - 3 b , D 1 =30+3 b ,and D 2 = 12 + 15 = 27. Thus x = D 1 /D =(30+3 b ) / (24 - 3 b )=(10+ b ) / (8 - b ), y = D 2 /D =27 / (24 - 3 b )=9 / (8 - b ). If D = 0 we cannot divide by D to obtain a solution. D =0if b =8 . T6.3-2 The session is ± A = [2,1,2;0,3,1;2,-3,4]; ± D2 = det([A(:,1),[17,6,19] 0 ,A(:,3)]); ± x = D2/det(A) y= 1 T6.4-1 The Matlab session is ± A = [3,12,-7;5,-6,-5;-2,7,9]; ± b = [5;-8;5]; ± rank(A) ans = 3 ± rank([A b]) ans = 3 ± x = A\b -1.0204 0.5940 -0.1332 ± A*x ans = 5.0000 -8.0000 5.0000 6-2
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These numbers are the right hand side of the equations. Thus the answer x = - 1 . 0204, y =0 . 5940, z = - 0 . 1332 is correct. Because the ranks are 3 and equal the number of unknowns, the solution is unique. T6.4-2 The Matlab session is ± A = [1,3,2;1,1,1]; ± b = [2;4]; ± rank(A) ans = 2 ± rank([A b]) ans = 2 ± pinv(A)*b ans = 4.3333 -1.6667 1.3333 ± x = A\b x= 5.0000 -1.0000 0 Because the ranks are 2 and less than the number of unknowns, the solution is not unique. The minimum-norm solution is x =4 . 3333, y = - 1 . 6667, z =1 . 333. The left-division solution is x =5 , y = - 1, z . T6.4-3 The Matlab session is ± A = [3,5,6;8,-1,2;5,-6,-4]; ± b = [6;1;-5]; ± rank(A) ans = 2 ± rank([A b]) ans = 2 ± rref([A b]) ans = 1.0000 0 0.3721 0.2558 0 1.0000 0.9767 1.0465 0000 6-3
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± x = pinv(A)*b x= 0.0571 0.5249 0.5340 ± x = A\b Warning: Matrix is singular to working precision. Inf Inf Inf Because the ranks are less than the number of unknowns, an infinite number of solutions exists. The reduced row echelon form gives the following solutions: x = - 0 . 3721 z +0 . 2558 y = - 0 . 9767 z +1 . 0465 where z can have any value. The pseudoinverse solution is x =0 . 0571, y . 5249, z . 534. T6.4-4 The Matlab session is ± A = [3,5,6;8,-1,2;5,-6,-4]; ± b = [6;1;-5]; ± rank(A) ans = 2 ± rank([A b]) ans = 2 rref([A b]) ans = 1.0000 0 -0.2727 5.2727 0 1.0000 1.3636 -2.3636 ± x = pinv(A)*b 4.8394 -0.1972 -1.5887 ± x = A\b 4.8000 6-4
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0 -1.7333 Because the ranks are equal but less than the number of unknowns, an infinite number of solutions exists. These are given by the reduced row echelon form: x =0 . 2727 z +5 . 2727 y = - 1 . 3636 z - 2 . 3636 where z can have any value.
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palm_ch6 - Solutions to Problems in Chapter Six Test Your...

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