Explanation Given Information: The provided matrix is ( I − A ) − 1 = [ 1.5 0.1 0 0.1 1.1 0.1 0 0 1.3 ] . Formula used: When a matrix has m number of rows and n number of 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 . Calculation: Consider the provided matrix ( I − A ) − 1 = [ 1.5 0.1 0 0.1 1.1 0.1 0 0 1.3 ] , As, the change in production for each sector 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. Since change in demand is increase in demand of 1 unit in each sector. Write change in demand as a column vector. Thus, change in demand is D + = [ 1 1 1 ] . Substitute [ 1.5 0.1 0 0.1 1.1 0.1 0 0 1.3 ] for ( I − A ) − 1 and [ 1 1 1 ] for D + in ( I − A ) − 1 D + . ( I − A ) − 1 D + = [ 1.5 0.1 0 0.1 1.1 0.1 0 0 1.3 ] [ 1 1 1 ] 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 the matrix [ 1.5 0.1 0 0.1 1.1 0.1 0 0 1.3 ] is 3 and number of columns is 3 , Substitute 3 for m and 3 for n in m × n , Thus, dimension of the matrix [ 1.5 0.1 0 0.1 1.1 0.1 0 0 1.3 ] is 3 × 3 . Since, number of rows in the matrix [ 1 1 1 ] is 3 and number of columns is 1 , Substitute 3 for m and 1 for n in m × n , Thus, dimension of the matrix [ 1 1 1 ] is 3 × 1

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Chapter 4.5, Problem 16E

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To calculate: The increase in production required to meet an increase in demand of 1 units in each Sector and what does columns of the matrix represent (I−A)−1 where (I−A)−1=[1.50.100.11.10.1001.3].

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The increase in production required to meet an increase in demand of 1 units in each Sector and what does columns of the matrix represent ( I − A ) − 1 where ( I − A ) − 1 = [ 1.5 0.1 0 0.1 1.1 0.1 0 0 1.3 ] .

Solution Summary: The author calculates the change in production required to meet an increase in demand of 1 units in each Sector.

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To calculate: The increase in production required to meet an increase in demand of 1 units in each Sector and what does columns of the matrix represent (I−A)−1 where (I−A)−1=[1.50.100.11.10.1001.3].

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