Explanation Given: The given matrix is: A = [ 0 0 1 0 1 2 0 0 1 ] Formula used: Compute the characteristics polynomial p ( r ) = | A − r I | . Find the zeroes of p ( r ) and express it as p ( r ) = ( r 1 − r ) m 1 ( r 2 − r ) m 2 ⋯ ( r k − r ) m k here r 1 , r 2 , ⋯ r k are the eigenvalues and m 1 , m 2 ⋯ , m k are their multiplicities. For each eigenvalue r i find m i linearly independent generalized eigenvectors by applying the Gauss-Jordan algorithm to the system ( A − r i I ) m i u = 0 . Then, assemble the fundamental matrix X ( t ) and find the general solution: x ( t ) = X 1 ( t ) + X 2 ( t ) + X 3 ( t ) Calculation: Consider the given matrix: A = [ 0 0 1 0 1 2 0 0 1 ] The characteristics polynomial is: | [ 0 0 1 0 1 2 0 0 1 ] − r [ 1 0 0 0 1 0 0 0 1 ] | = 0 | [ 0 0 1 0 1 2 0 0 1 ] − [ r 0 0 0 r 0 0 0 r ] | = 0 | [ − r 0 1 0 1 − r 2 0 0 1 − r ] | = 0 − r ⋅ [ 1 − r 2 0 1 − r ] + 1 ⋅ [ 0 1 − r 0 0 ] = 0 Further simplify as: − r [ ( 1 − r ) 2 ] = 0 r = 0 , 1 , 1 The eigenvalues are 0 and 1 with multiplicity 2. The eigenvector corresponding to the eigenvalue r = 0 : ( [ 0 0 1 0 1 2 0 0 1 ] − 0 [ 1 0 0 0 1 0 0 0 1 ] ) [ u 1 u 2 u 3 ] = [ 0 0 0 ] ( [ 0 0 1 0 1 2 0 0 1 ] − [ 0 0 0 0 0 0 0 0 0 ] ) [ u 1 u 2 u 3 ] = [ 0 0 0 ] [ 0 0 1 0 1 2 0 0 1 ] [ u 1 u 2 u 3 ] = [ 0 0 0 ] It indicates as, u 3 = 0 , u 1 is free and u 2 + 2 u 3 = 0

Fundamentals Of Differential Equations And Boundary Value Problems Plus Mylab Math With Pearson Etext -- Title-specific Access Card Package (7th ... Fundamentals Of Differential Equations)

7th Edition

ISBN: 9780134768717

Author: R. Kent Nagle, Edward B. Saff, Arthur David Snider

Publisher: PEARSON

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Chapter 9.8, Problem 18E

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Chapter 9.8, Problem 17E

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5) State any theorems that you use in determining your solution. a) Suppose you are given a model with two explanatory variables such that: Yi = a +ẞ1x1 + ẞ2x2i + Ui, i = 1, 2, ... n Using partial differentiation derive expressions for the intercept and slope coefficients for the model above. [25 marks] b) A production function is specified as: Yi = α + B₁x1i + ẞ2x2i + Ui, i = 1, 2, ... n, u₁~N(0,σ²) where: y = log(output), x₁ = log(labor input), x2 = log(capital input) The results are as follows: x₁ = 10, x2 = 5, ỹ = 12, S11 = 12, S12= 8, S22 = 12, S₁y = 10, = 8, Syy = 10, S2y n = 23 (individual firms) i) Compute values for the intercept, the slope coefficients and σ². [20 marks] ii) Show that SE (B₁) = 0.102. [15 marks] iii) Test the hypotheses: ẞ1 = 1 and B2 = 0, separately at the 5% significance level. You may take without calculation that SE (a) = 0.78 and SE (B2) = 0.102 [20 marks] iv) Find a 95% confidence interval for the estimate ẞ2. [20 marks]

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The general solution for the given matrix.