Substituting these equations into the expression for

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Unformatted text preview: e equations into the expression for the dot product we obtain an equation for t 0= = = = (x ? 2)2 + (y ? 1)(y ? 3) + (z + 1)(z ? 3) (5 ? 2t)2 + (?2 + t)(?4 + t) + (t + 1)(t ? 3) 6t2 ? 28t + 30 2(3t ? 5)(t ? 3) 5 with solutions t = 3 and t = 3 . Putting these back into the general solution above ? 5 yields the two possibilities B = (1; 2; 3) and B = 11 ; 2 ; 3 . Since the coordinates of B 33 are integers we must have B = (1; 2; 3). 3. In each of these questions the method is to write a matrix A whose columns are the vectors in question so that the span of the vectors is the row space of A. Then the vectors form a basis for their span if and only if row reducing A yields a matrix in which every column has a pivot. (a) The matrix A is 2 3 2 3 1 ?2 1 ?2 4 ?2 4 5 ?! 4 0 0 5 1 ?2 00 The second column doesn't have a pivot so the vectors do not form a basis for their span. (b) The matrix A is 2 3 2 3 23 10 4 5 1 5 ?! 4 0 1 5 11 00 Both columns have pivots so the vectors do form a basis for their span. (c) The matrix A is 2 3 2 3 2 3 ?1 10 1 45 1 4 5 ?! 4 0 1 ?1 5 11 0 00 0 The third column has no pivots so the vectors do not form a basis for their span. MATA04Y page 5 4. (a) The input-output matrix for this economy is 2 3 1=4 0 1=2 0 6 1=2 1=3 0 1=8 7 E = 6 1=6 2=3 1=2 1=4 7 6 7 4 5 1=12 0 0 5=8 where we have used the fact that the column sums are all 1 to ll in the diagonal elements. To determine the price structure for this economy we must solve E v = v, or, what is the same (E ? I )v = 0. Finding the reduced row echelon form for E ? I yields 21 2 3 1 93 ?1 0 0 1 0 0 ?2 4 2 61 6 0 1 0 ? 57 7 7 1 1 0 62 6 7 3 ?1 8 16 7 E ?I =6 1 7: 7 ?! 6 1 1 2 46 4 0 0 1 ? 27 5 5 3 2 ?1 4 4 1 000 0 0 0 5 ?1 12 8 Thus the equilibrium price structure is given by 2 6 6 4 p = t6 9 2 57 16 27 4 3 7 7 7 5 ; t2R : 1 (b) The input-output matrix has the pattern 2 0 0 6 0 6 00 Squaring this pattern yields the pattern 2 6 6 4 4 3 7 7 5 : 3 7 7 5 0 and squaring again yields 2 6 6 4 3 7 7 5 : Since no entry in the fourth power is zero, E is a regular stochastic matrix. MATA04Y 5. (a) We may carry out a seq...
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This note was uploaded on 04/13/2010 for the course ECON 330 taught by Professor Minetti during the Fall '08 term at Michigan State University.

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