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(4) For what values of z is the matrix A invertible? (5) Solve the system of equations in part (2) by Cramer's rule when z = 0.5.

(4) For what values of z is the matrix A invertible? (5) Solve the system of equations in part (2) by Cramer's rule when z = 0.5.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 28EQ
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(4) For what values of z is the matrix A invertible? (5) Solve the system of equations in part (2) by Cramer's rule when z = 0.5.
Transcribed Image Text:(4) For what values of z is the matrix A invertible? (5) Solve the system of equations in part (2) by Cramer's rule when z = 0.5.
Question 2 We assume that an economy has only three sectors: Goods, Services and Consumption. The Goods sector has a total annual output of r units, the Services sector has a total annual output of y units, and the Consumption sector does not produce goods or services but only consumes 18 units of Goods and 11 units of Services annually. Each year, to meet the consumer demand, for its own production: (i) the Goods sector must use 10% of its own total output and 60% of the total output from the Services sector; and (ii) the Services sector must use 100z% pereent of its own total output and 50% of the total output from the Goods sector, where z is an unknown real number between 0 and 1. The economy is in equilibrium if for each production sector (of Goods and Services), the output is equal to the sum of its input and the consumer demand. (1) Derive a two-equation linear system to demonstrate the relationship between z and y, treating z as an unknown constant. (2) Re-express the above linear system in a matrix form: A Show all the elements of A and b. (3) What is the determinant of matrix A in terms of z (that is, as a function of z)?
Transcribed Image Text:Question 2 We assume that an economy has only three sectors: Goods, Services and Consumption. The Goods sector has a total annual output of r units, the Services sector has a total annual output of y units, and the Consumption sector does not produce goods or services but only consumes 18 units of Goods and 11 units of Services annually. Each year, to meet the consumer demand, for its own production: (i) the Goods sector must use 10% of its own total output and 60% of the total output from the Services sector; and (ii) the Services sector must use 100z% pereent of its own total output and 50% of the total output from the Goods sector, where z is an unknown real number between 0 and 1. The economy is in equilibrium if for each production sector (of Goods and Services), the output is equal to the sum of its input and the consumer demand. (1) Derive a two-equation linear system to demonstrate the relationship between z and y, treating z as an unknown constant. (2) Re-express the above linear system in a matrix form: A Show all the elements of A and b. (3) What is the determinant of matrix A in terms of z (that is, as a function of z)?
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