**Problem Statement:**Find the production schedule for the technology matrix and demand vector given below:\[A = \begin{bmatrix} 0.4 & 0.1 & 0 \\ 0.4 & 0.6 & 0.8 \\0.1 & 0.3 & 0.2 \end{bmatrix}, \quad D = \begin{bmatrix} 5 \\ 8 \\ 3 \end{bmatrix}\]\[X = \begin{bmatrix} □ \\ □ \\ □ \end{bmatrix}\]**Explanation:**- The matrix $ A $ is the technology matrix that represents the input-output coefficients for a production model. It is a 3x3 matrix where each element describes the proportion of resources from one sector needed to produce a unit of output in another sector.- The vector $ D $ is the demand vector, which indicates the external demand for products from each sector. It is a 3x1 matrix.- The vector $ X $ represents the production schedule needed to meet the external demand and internal technological requirements. It is the solution to the equation $ X = (I - A)^{-1} D $, where $ I $ is the identity matrix.

The given technology matrix is A=0.40.100.40.60.80.10.30.2 and demand vector is D=583. Let x=x1x2x3 be the production schedule.Using the equation I-AX=D, find the production schedule as follows. I-AX=D100010001-0.40.100.40.60.80.10.30.2x1x2x3=5830.6-0.10-0.4-0.40.8-0.1-0.30.8x1x2x3=583 The augmented matrix of the above system is 0.6-0.105-0.40.4-0.88-0.1-0.30.83.Find the reduced row echelon form of 0.6-0.105-0.40.4-0.88-0.1-0.30.83 as shown below. 0.6-0.105-0.40.4-0.88-0.1-0.30.83~1-160253-0.40.4-0.88-0.1-0.30.83R1→0.6R1~1-160253013-45343-0.1-0.30.83R2→R2+0.4R1~1-160253013-453430-196045236R3→R3+0.1R1~1-16025301-125340-196045236R2→3R2~10-251401-125340-196045236R1→R1+16R2~10-251401-1253400125735R3→R3+1960R2~10016001-1253400125735R1→R1+10R3~10016001091000125735R2→R2+60R3~100160010910001365R3→25R3Thus, the reduced row echelon form of 0.6-0.105-0.40.4-0.88-0.1-0.30.83 is 100160010910001365. The linear system corresponding to the reduced row echelon form 100160010910001365 is, x1=160x2=910x3=365 Therefore, the production schedule is x=160910365.

Answered: Find the production schedule for the…

Math

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Find the production schedule for the technology matrix and demand vector given below: 0.4 0.1 5 A = 0.4 0.6 0.8 D 8 0.1 0.3 0.2 3 X =

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

**Problem Statement:**

Find the production schedule for the technology matrix and demand vector given below:

\[
A = \begin{bmatrix}
0.4 & 0.1 & 0 \\
0.4 & 0.6 & 0.8 \\
0.1 & 0.3 & 0.2
\end{bmatrix}, \quad
D = \begin{bmatrix}
5 \\
8 \\
3
\end{bmatrix}
\]

\[
X = \begin{bmatrix}
□ \\
□ \\
□
\end{bmatrix}
\]

**Explanation:**

- The matrix $ A $ is the technology matrix that represents the input-output coefficients for a production model. It is a 3x3 matrix where each element describes the proportion of resources from one sector needed to produce a unit of output in another sector.

- The vector $ D $ is the demand vector, which indicates the external demand for products from each sector. It is a 3x1 matrix.

- The vector $ X $ represents the production schedule needed to meet the external demand and internal technological requirements. It is the solution to the equation $ X = (I - A)^{-1} D $, where $ I $ is the identity matrix.

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