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Chapter 2: Problem 3

Use the input-output matrix \(A\) and the consumer demand matrix \(D\) to solve the matrix equation \((I-A) X=D\) for the total output matrix \(X\) $$ A=\left[\begin{array}{ll} 0.4 & 0.2 \\ 0.3 & 0.1 \end{array}\right] \text { and } D=\left[\begin{array}{l} 10 \\ 12 \end{array}\right] $$

Short Answer

Expert verified
The total output matrix \(X\) is found by solving the matrix equation \((I-A)X=D\) using the provided input-output matrix \(A\) and consumer demand matrix \(D\). After calculating \((I-A)^{-1}\) and multiplying it with \(D\), we get the total output matrix \(X = \left[\begin{array}{c} 23.75 \\ 21.25 \end{array}\right]\).

Step by step solution

01

Find (I-A)

Here, I is the identity matrix of the same order as A, which is a 2x2 matrix. So, I = \(\left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right]\) Now, subtract A from I: \(I - A = \left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right] - \left[\begin{array}{cc} 0.4 & 0.2 \\ 0.3 & 0.1 \end{array}\right] = \left[\begin{array}{cc} 0.6 & -0.2 \\ -0.3 & 0.9 \end{array}\right]\)
02

Calculate the inverse of (I-A)

Calculate the inverse of the matrix (I-A), denoted as \((I-A)^{-1}\). The formula for the inverse of a 2x2 matrix \(\left[\begin{array}{cc} a & b \\ c & d \end{array}\right]\) is \(\frac{1}{ad-bc}\left[\begin{array}{cc} d & -b \\ -c & a \end{array}\right]\) Apply the formula to (I-A): \((I-A)^{-1} = \frac{1}{(0.6)(0.9) - (-0.2)(-0.3)}\left[\begin{array}{cc} 0.9 & 0.2 \\ 0.3 & 0.6 \end{array}\right] = \frac{1}{0.48}\left[\begin{array}{cc} 0.9 & 0.2 \\ 0.3 & 0.6 \end{array}\right] = \left[\begin{array}{cc} 1.875 & 0.41667 \\ 0.625 & 1.25 \end{array}\right]\)
03

Multiply (I-A)^{-1} with D to get X

Multiply \((I-A)^{-1}\) with D to find the total output matrix X: \(X = (I-A)^{-1}D = \left[\begin{array}{cc} 1.875 & 0.41667 \\ 0.625 & 1.25 \end{array}\right]\left[\begin{array}{c} 10 \\ 12 \end{array}\right]\) Using matrix multiplication: \(X = \left[\begin{array}{cc} 1.875(10) + 0.41667(12) \\ 0.625(10) + 1.25(12) \end{array}\right] = \left[\begin{array}{cc} 18.75 + 5 \\ 6.25 + 15 \end{array}\right] = \left[\begin{array}{c} 23.75 \\ 21.25 \end{array}\right]\) So, the total output matrix X is: \(X = \left[\begin{array}{c} 23.75 \\ 21.25 \end{array}\right]\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Equations
Matrix equations, much like regular equations, represent a series of operations. However, instead of dealing with single values, matrices involve arrays of numbers. This can be used to model complex systems like economic input-output relationships.
The general form of a matrix equation is often denoted as \(AX = B\), where:
  • \(A\) is a matrix containing known coefficients
  • \(X\) represents an unknown matrix to solve for
  • \(B\) is the result matrix
In our exercise, the equation \((I-A)X=D\) is used, where \((I-A)\) operates as the coefficient matrix reflecting changes in the economy, \(X\) is the total output matrix, and \(D\) is the consumer demand matrix.
Inverse Matrix
An inverse matrix is pivotal when solving equations like \((I-A)X=D\). The inverse of a matrix \(A\), denoted as \(A^{-1}\), satisfies the equation \(AA^{-1} = I\), where \(I\) is the identity matrix. This inverse allows us to "undo" the effect of matrix \(A\) and solve for unknowns.

To calculate the inverse of a 2x2 matrix:
  • The matrix's determinant \(ad-bc\) must be non-zero
  • Apply the formula: \(\frac{1}{ad-bc}\left[\begin{array}{cc} d & -b \ -c & a \end{array}\right]\)
Using this formula simplifies our task of finding \((I-A)^{-1}\) in the problem, making it possible to isolate and solve for the total output matrix \(X\).
Matrix Multiplication
Matrix multiplication is essential for finding solutions to matrix equations like our \(X = (I-A)^{-1}D\). It follows specific rules different from normal multiplication.
In matrix multiplication, the "dot product" of the rows and columns must be calculated:
  • The number of columns in the first matrix must match the number of rows in the second
  • For each element, multiply corresponding elements and add them
Through matrix multiplication, the inverse matrix \((I-A)^{-1}\) operates on demand \(D\) to yield the output \(X\). Remember, the order of multiplication matters; \(AB\) is not the same as \(BA\).
Identity Matrix
The identity matrix is a special matrix that behaves like the number 1 in multiplication. When any matrix \(A\) is multiplied by the identity matrix \(I\), the result is \(A\) itself. This property is crucial for understanding inverse matrices.
In a \(2x2\) identity matrix:
  • The diagonal from top left to bottom right is filled with 1s
  • All other elements are 0s
For example, the \(2x2\) identity matrix is \(\left[\begin{array}{cc} 1 & 0 \ 0 & 1 \end{array}\right]\). Because it maintains matrix \(A\)'s form when multiplied by \(A^{-1}\), it confirms that \(A\) and \(A^{-1}\) are true inverses. In our exercise, calculating \((I-A)\) begins with understanding the structure of \(I\).

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Most popular questions from this chapter

Three network consultants, Alan, Maria, and Steven, each received a year-end bonus of \(\$ 10,000\), which they decided to invest in a \(401(\mathrm{k})\) retirement plan sponsored by their employer. Under this plan, employees are allowed to place their investments in three funds: an equity index fund (I), a growth fund (II), and a global equity fund (III). The allocations of the investments (in dollars) of the three employees at the beginning of the year are summarized in the matrix $$ \begin{array}{l} \text { II }\\\ \begin{array}{c} \text { Alan } \\ A=\text { Maria } \\ \text { Steven } \end{array}\left[\begin{array}{lll} 4000 & 3000 & 3000 \\ 2000 & 5000 & 3000 \\ 2000 & 3000 & 5000 \end{array}\right] \end{array} $$ The returns of the three funds after 1 yr are given in the matrix $$ \begin{array}{r} \mathrm{I} \\ B=\mathrm{II} \\ \mathrm{III} \end{array}\left[\begin{array}{l} 0.18 \\ 0.24 \\ 0.12 \end{array}\right] $$ Which employee realized the best return on his or her investment for the year in question? The worst return?

Mr. and Mrs. Garcia have a total of $$\$ 100,000$$ to be invested in stocks, bonds, and a money market account. The stocks have a rate of return of \(12 \% /\) year, while the bonds and the money market account pay \(8 \% /\) year and \(4 \%\) year, respectively. The Garcias have stipulated that the amount invested in the money market account should be equal to the sum of \(20 \%\) of the amount invested in stocks and \(10 \%\) of the amount invested in bonds. How should the Garcias allocate their resources if they require an annual income of $$\$$ 10,000$ from their investments?

Matrix \(A\) is an input-output matrix associated with an economy, and matrix \(D\) (units in millions of dollars) is a demand vector. In each problem,find the final outputs of each industry such that the demands of industry and the consumer sector are met. $$ A=\left[\begin{array}{lll} 0.2 & 0.4 & 0.1 \\ 0.3 & 0.2 & 0.1 \\ 0.1 & 0.2 & 0.2 \end{array}\right] \text { and } D=\left[\begin{array}{r} 6 \\ 8 \\ 10 \end{array}\right] $$

The property damage claim frequencies per 100 cars in Massachusetts in the years 2000,2001 , and 2002 are \(6.88,7.05\), and \(7.18\), respectively. The corresponding claim frequencies in the United States are 4.13, \(4.09\), and \(4.06\), respectively. Express this information using a \(2 \times 3\) matrix.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A^{-1}\) does not exist, then the system \(A X=B\) of \(n\) linear equations in \(n\) unknowns does not have a unique solution.
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