Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Educational Exercise on Input Coefficients Matrix
**Problem Statement:**
Consider the following input coefficients matrix:
\[
A = \begin{pmatrix}
0.25 & 0.5 & 0.2 \\
0 & 0.1 & 0 \\
0 & 0.1 & 0.3
\end{pmatrix}
\]
**Tasks:**
(i) Find \((I-A)^{-1}\).
(ii) Hence, find the vector inputs required to produce the final demand vector:
\[
\begin{pmatrix}
0 \\
6 \\
3
\end{pmatrix}
\]
(iii) Find the primary inputs coefficient vector (slide 14, lecture 13).
(iv) Suppose labour is the sole primary input vector and there are 8 available units (in value) of labour. Is that sufficient to produce the final demand vector in (ii)?
---
This exercise involves understanding and computing various transformations of an input coefficients matrix. The matrix represents interdependencies between different sectors or entities, with coefficients denoting the proportion of one entity’s output required to meet the demands of another.
**Explanation:**
- Task (i) involves calculating the inverse of the matrix \( (I-A) \), where \( I \) is the identity matrix. This step is crucial for understanding how changes in one sector affect others.
- Task (ii) requires using the inverted matrix to determine the vector inputs necessary to achieve a specified final output or demand.
- Task (iii) involves identifying the primary inputs, which are fundamental resources or factors needed in initial stages of production.
- Task (iv) examines whether the available resources (labour) are enough to meet the demands, applying practical constraints to theoretical computations.
This problem supports learning in matrix algebra and economic input-output analysis, important for students studying economics or quantitative methods in business.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27d05195-d4e5-49f1-8dc4-61bbc8dbf8a7%2F27cd5238-e0d8-4df6-9797-1361a169817a%2Fr0thlfbk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Exercise on Input Coefficients Matrix
**Problem Statement:**
Consider the following input coefficients matrix:
\[
A = \begin{pmatrix}
0.25 & 0.5 & 0.2 \\
0 & 0.1 & 0 \\
0 & 0.1 & 0.3
\end{pmatrix}
\]
**Tasks:**
(i) Find \((I-A)^{-1}\).
(ii) Hence, find the vector inputs required to produce the final demand vector:
\[
\begin{pmatrix}
0 \\
6 \\
3
\end{pmatrix}
\]
(iii) Find the primary inputs coefficient vector (slide 14, lecture 13).
(iv) Suppose labour is the sole primary input vector and there are 8 available units (in value) of labour. Is that sufficient to produce the final demand vector in (ii)?
---
This exercise involves understanding and computing various transformations of an input coefficients matrix. The matrix represents interdependencies between different sectors or entities, with coefficients denoting the proportion of one entity’s output required to meet the demands of another.
**Explanation:**
- Task (i) involves calculating the inverse of the matrix \( (I-A) \), where \( I \) is the identity matrix. This step is crucial for understanding how changes in one sector affect others.
- Task (ii) requires using the inverted matrix to determine the vector inputs necessary to achieve a specified final output or demand.
- Task (iii) involves identifying the primary inputs, which are fundamental resources or factors needed in initial stages of production.
- Task (iv) examines whether the available resources (labour) are enough to meet the demands, applying practical constraints to theoretical computations.
This problem supports learning in matrix algebra and economic input-output analysis, important for students studying economics or quantitative methods in business.
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