Explanation Given Information: An external demand for wood products rises by $ 10 , 000 million and external demand for paper and allied products rises by $ 20 , 000 million and the input output table is Wood Paper Wood 36,000 7,000 Paper 100 17,000 Total Output 120,000 120,000 Formula used: When a matrix has m rows and n columns, then the matrix is said to have dimension of m × n . For two matrices A and B with equal dimension, the sum of difference is simplified as, ( a 11 a 12 a 21 a 22 ) ± ( b 11 b 12 b 21 b 22 ) = ( a 11 ± b 11 a 12 ± b 12 a 21 ± b 21 a 22 ± b 22 ) When a matrix has m rows and n columns, then the matrix is said to have dimension of m × n . According to the product rule of matrices, the matrix X with dimension m × n and the matrix Y with dimension n × k , the product XY is the matrix of dimension m × k and i j t h entry of AB is the sum of product of corresponding entries of row i of X and column j of Y . According to the property of inverse, the inverse of a 2 × 2 matrix A = [ a b c d ] is A − 1 = 1 a d − b c [ d − b − c a ] The variables a , b , c and d in the matrix are arbitrary numbers. Divide each column of the input-output matrix by the total of the matrix. Calculation: Consider external demand for wood products rises by $ 10 , 000 million and external demand for paper and allied products rises by $ 20 , 000 million and the input output table is Wood Paper Wood 36,000 7,000 Paper 100 17,000 Total Output 120,000 120,000 In order to calculate the technology matrix, divide each column of the input-output by its total. Write the technology matrix from the table. A = [ 36 , 000 120 , 000 7 , 000 120 , 000 100 120 , 000 17 , 000 120 , 000 ] = [ 3 10 7 120 1 1200 17 120 ] Thus, technology matrix is A = [ 3 10 7 120 1 1200 17 120 ] . Since, an external demand for $ 80 , 000 millions of communication equipment and $ 90 , 000 millions of electronic components. Write the demand vector. D = [ 10 , 000 20 , 000 ] Recall that the production vector X satisfies X = ( I − A ) − 1 D , where A is the technology matrix, D is the demand vector and I is the identity matrix. Substitute [ 3 10 7 120 1 1200 17 120 ] for A , [ 1 0 0 1 ] for I , and [ 10 , 000 20 , 000 ] for D in X = ( I − A ) − 1 D . X = ( I − A ) − 1 D = ( [ 1 0 0 1 ] − [ 3 10 7 120 1 1200 17 120 ] ) − 1 [ 10 , 000 20 , 000 ] As the matrix with m rows and n columns is of dimension m × n , where m and n are positive integers

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ISBN: 9781337280426

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 4.5, Problem 22E

To determine

To calculate: The increase in production of the two sectors one for lumber and wood products and the other for paper and allied products where an external demand for wood products rises by $10,000 million and external demand for paper and allied products rises by $20,000 million and the input output table is

Wood Paper

Wood 36,000 7,000

Paper 100 17,000

Total Output 120,000 120,000

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Chapter 4.5, Problem 21E

Chapter 4.5, Problem 23E

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	Wood	Paper
Wood	36,000	7,000
Paper	100	17,000
Total Output	120,000	120,000

Solution Summary: The author explains that the increase in production of lumber and wood products and paper and allied products is 16,000 and the sum of difference is simplified for two matrices A and B with equal dimension.

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