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Chapter 2: Q2.6-10Q (page 93)

The consumption matrix C for the U.S. economy in 1972 has the property that every entryin the matrix \({\left( {I - C} \right)^{ - 1}}\) is nonzero (and positive). What does that say about the effect of raising the demand for the output of just one sector of the economy?

Short Answer

Expert verified

An increase in the final demand for any sector will increase the demand for all sectors.

Step by step solution

01

State the \(\left( {i,j} \right)\) entry in \({\left( {I - C} \right)^{ - 1}}\)

From Exercise 8, the \(\left( {i,j} \right)\) entry in \({\left( {I - C} \right)^{ - 1}}\) corresponds to the effect on the production of sector \(i\) if the final demand for the output of sector \(j\) rises by one unit.

02

Find the effect of raising the demand for the output of one sector

An increase in the final demand for any sector increases the production of all sectors because all the entries are positive.

Therefore, an increase in the final demand for any sector increases the demand for all sectors.

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Most popular questions from this chapter

Let T be a linear transformation that maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\). Is \({T^{ - 1}}\) also one-to-one?

The solution to the steady-state heat flow problem for
the plate in the figure is approximated by the solution to the
equation\(A{\bf{x}} = {\bf{b}}\);where\(b = \left( {5,15,0,10,0,10,20,30} \right)\)and

\(A = \left[ {\begin{array}{*{20}{c}}4&{ - 1}&{ - 1}&{}&{}&{}&{}&{}\\{ - 1}&4&0&{ - 1}&{}&{}&{}&{}\\{ - 1}&0&4&{ - 1}&{ - 1}&{}&{}&{}\\{}&{ - 1}&{ - 1}&4&0&{ - 1}&{}&{}\\{}&{}&{ - 1}&0&4&{ - 1}&{ - 1}&{}\\{}&{}&{}&{ - 1}&{ - 1}&4&0&{ - 1}\\{}&{}&{}&{}&{ - 1}&0&4&{ - 1}\\{}&{}&{}&{}&{}&{ - 1}&{ - 1}&4\end{array}} \right]\)

(Refer to Exercise 33 of Section 1.1.) The missing entries in Aare zeros. The nonzero entries of A lie within a band along the main diagonal. Such band matricesoccur in a variety of applications and often are extremely large (with thousands of rows and columns but relatively narrow bands).

  1. Use the method of Example 2 to construct an LU factorization of A, and note that both factors are band matrices (with two nonzero diagonals below or above the main diagonal). Compute \(LU - A\) to check your work.
  1. Use the LU factorization to solve\(A{\bf{x}} = {\bf{b}}\).
  1. Obtain \({A^{ - {\bf{1}}}}\) and note that\({A^{ - {\bf{1}}}}\) is a dense matrix with no
    band structure. When Ais large, LandUcan be stored in
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1. Find the inverse of the matrix \(\left( {\begin{aligned}{*{20}{c}}{\bf{8}}&{\bf{6}}\\{\bf{5}}&{\bf{4}}\end{aligned}} \right)\).

Suppose columns 1, 3, 5, and 6 of a matrix A are linearly independent (but are not necessarily pivot columns) and the rank of A is 4. Explain why the four columns mentioned must be a basis for the column space of A.

In Exercise 11, give integers p and q such that Nul A is a subspace of \[{\mathbb{R}^p}\] and Col A is a subspace of \[{\mathbb{R}^q}\].

11. \[A = \left[ {\begin{array}{*{20}{c}}{\bf{3}}&{\bf{2}}&{\bf{1}}&{ - {\bf{5}}}\\{ - {\bf{9}}}&{ - {\bf{4}}}&{\bf{1}}&{\bf{7}}\\{\bf{9}}&{\bf{2}}&{ - {\bf{5}}}&{\bf{1}}\end{array}} \right]\]
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