Explanation Given Information: The technology matrix is A = [ 0.7 0.1 0.1 0.7 ] and external demand vector is D = [ 1 , 000 2 , 000 ] . Formula used: The production vector is ( I − A ) − 1 D where D external demand vector, A is the technology matrix and I is the identity matrix. A matrix with m rows and n columns is of dimension m × n , where m and n are positive integers. For two matrices A and B with equal dimension, A + B and A − B is the sum and difference of the corresponding entries of the matrices respectively. For matrices A with dimension m × n and B with dimension k × l , the product AB is defined if the number of columns in A equal number of columns in B , n = k . For matrices A with dimension m × n and B with dimension n × k , the product AB is the matrix of dimension m × k and ij th entry of AB is the sum of product of corresponding entries of row i of A and column j of B . Inverse of a 2 × 2 matrix [ a b c d ] is 1 a d − b c [ d − b − c a ] where a , b , c and d are arbitrary numbers. Calculation: Consider technology matrix is A = [ 0.7 0.1 0.1 0.7 ] and external demand vector is D = [ 1 , 000 2 , 000 ] . Recall that the production vector is ( I − A ) − 1 D where D is the change in external demand vector, A is the technology matrix and I is the identity matrix. Substitute [ 0.7 0.1 0.1 0.7 ] for A , [ 1 0 0 1 ] for I , and [ 1 , 000 2 , 000 ] for D in ( I − A ) − 1 D . ( I − A ) − 1 D = ( [ 1 0 0 1 ] − [ 0.7 0.1 0.1 0.7 ] ) − 1 × [ 1 , 000 2 , 000 ] Recall that a matrix with m rows and n columns is of dimension m × n , where m and n are positive integers. Since, number of rows in [ 0.7 0.1 0.1 0.7 ] is 2 and number of columns is 2 , Substitute 2 for m and 2 for n in m × n . Thus, dimension of [ 0.7 0.1 0.1 0.7 ] is 2 × 2 . Since, number of rows in [ 1 0 0 1 ] is 2 and number of columns is 2 , Substitute 2 for m and 2 for n in m × n . Thus, dimension of [ 1 0 0 1 ] is 2 × 2 . Thus, both the matrices have same dimensions. Since, number of rows in [ 1 , 000 2 , 000 ] is 2 and number of columns is 1 , Substitute 2 for m and 1 for n in m × n . Thus, dimension of [ 1 , 000 2 , 000 ] is 2 × 1 . Recall that for two matrices A and B with equal dimension, A − B is the and difference of the corresponding entries of the matrices. Simplify [ 1 0 0 1 ] − [ 0.7 0.1 0.1 0.7 ] . [ 1 0 0 1 ] − [ 0.7 0.1 0.1 0.7 ] = [ 0.3 − 0.1 − 0.1 0

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Author: Stefan Waner, Steven Costenoble

Publisher: Brooks Cole

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Chapter 4, Problem 26RE

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To calculate: The production vector where the technology matrix is A=[0.70.10.10.7] and external demand vector is D=[1,0002,000].

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Chapter 4, Problem 25RE

Chapter 4, Problem 27RE

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The production vector where the technology matrix is A = [ 0.7 0.1 0.1 0.7 ] and external demand vector is D = [ 1 , 000 2 , 000 ] .

Solution Summary: The author calculates the production vector where the technology matrix is A=left[cc0.7& 0.10.1&

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To calculate: The production vector where the technology matrix is A=[0.70.10.10.7] and external demand vector is D=[1,0002,000].

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Chapter 4 Solutions