💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Watch this step-by-step video, matched to your homework problem.

Try Numerade Free for 30 Days

Like

Report

Consider the production model $\mathbf{x}=C \mathbf{x}+\mathbf{d}$ for an economy with two sectors, where$$C=\left[\begin{array}{cc}{.0} & {.5} \\ {.6} & {.2}\end{array}\right], \quad \mathbf{d}=\left[\begin{array}{c}{50} \\ {30}\end{array}\right]$$Use an inverse matrix to determine the production level necessary to satisfy the final demand.

$=\left[\begin{array}{l}{110} \\ {120}\end{array}\right]$

Algebra

Chapter 2

Matrix Algebra

Section 6

The Leontief Input–Output Model

Introduction to Matrices

Campbell University

Oregon State University

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

06:52

Production Level Show that…

04:18

Find the production matrix…

05:25

03:17

Given the demand curve $p=…

19:44

Consider an economy with t…

02:17

Use inverse matrices to fi…

02:05

The demand and supply curv…

04:01

04:20

Let $C$ be an $n \times n$…

In this example, we are dealing with an economy that has the following consumption matrix C divide defined here as well as the demand vector d as here. What we like to do is to find out the production level which will call X given this matrix and this demand vector well, the first step is to four mount the model for this type of situation which is of the form X. The production level equals the intermediate demand, which is C Times X plus the final demand, which is deep Now if we rearrange this equation by subtracting, see extra bull sides, we have the equation I minus C times a vector X equals the demand vector d And then we know by our theorem in this section that this matrix is in vertebral. Since the column sums are both less than one. This then tells us that the solution will be X equals I minus C in verse times d. So let's start off then in our solution by finding I minus c inverse here. So we'll have I minus c is going to be first. I consist of just ones on the main diagonal, So we're going to subtract one minus zero from the main Dagnall of sea to produce a one here and will have a negative 0.5 here, a negative 0.6. Then take one minus the point to, and we have a 0.8 altogether. But what we need here is its inverse. So let's take the determinant of I minus C. It's going to be the product of the main diagonal. So 0.8 subtract the product of the off diagonal, which is negative 0.6 times 0.5, resulting in a point 30 And we get all together that the determinant of this matrix is 300.5, which also tells us it is in vertical as we predicted. So let's calculate its inverse Next I minus see in verse is going to be first take the main diagonal which is in purple and alternate it so point it goes here and 0.0.1 goes here. We're going to then negates the off diagonal these entries here So we have a positive 0.6 a positive 0.5. Then divide by the determinant which is one over 10.5 Well, one over 10.5 results in two, so the inverse becomes 1.61 1.2 and there was an issue right here. This should have been a positive one or a point not 10.0.1, but just one itself multiplied by two and we get a two here. So this is our inverse matrix. This now tells us that the production level required to meet this demand vector will be X equals First, the inverse matrix of I minus C, which is 1.6 1.212 Multiply by the demand vector D, which says take 50 from the first sector, 30 from the second sector as the demand and multiply we see then that will require ah 110 units from the first sector of the economy and 120 units from the second sector and the solves our problem here.

View More Answers From This Book

Find Another Textbook

Numerade Educator

In mathematics, the absolute value or modulus |x| of a real number x is its …

Production Level Show that if $r(x)=6 x$ and $c(x)=$$x^{3}-6 x^{2}+15 x$…

Find the production matrix for the following input-output and demand matrice…

Given the demand curve $p=35-q^{2}$ and the supply curve $p=3+q^{2},$ find t…

Consider an economy with three sectors, Chemicals \& Metals, Fuels \&…

Use inverse matrices to find the equilibrium point for the demand and supply…

The demand and supply curves for a product are given as$$ \begin{aligned…

Let $C$ be an $n \times n$ consumption matrix whose column sums are less tha…

03:15

Find an LU factorization of the matrices in Exercises $7-16$ (with $L$ unit …

05:06

Unless otherwise specified, assume that all matrices in these exercises are …

02:43

Suppose $A$ is $n \times n$ and the equation $A \mathbf{x}=\mathbf{0}$ has o…

00:33

Each equation in Exercises $1-4$ illustrates a property of determinants. Sta…

02:02

Solve the equation $A B=B C$ for $A,$ assuming that $A, B,$ and $C$ are squa…

02:03

Use Exercises $25-28$ to answer the questions in Exercises 31 and $32 .$ Giv…

05:17

Suppose $A D=I_{m}$ (the $m \times m$ identity matrix). Show that for any $\…

02:58

Explain why the columns of $A^{2}$ span $\mathbb{R}^{n}$ whenever the column…

00:39

02:09

$[\mathbf{M}]$ Find a column of the matrix in Exercise 39 that can be delete…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.