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Question One

A dietician in a hospital is to arrange a special diet for an injured athlete composing of two basic nutrients. The diet is to include 400 units of calcium and 780 units of iron. Two food sources are available. Each pound of food A contains 2 units of calcium and 5 units of iron; each pound of food B contains 6 units of calcium and 5 units of iron. How many pounds of each food source should be used to exactly meet the dietary requirement?
[3 mark]

Assume food B is no longer available, but a newly introduced food source, food C contains 3 units of calcium and 7 units of iron. Why would the grouping of food A and food C produce an inappropriate solution?
[3 mark] Solve by Gauss Jordan elimination and state the final solution.

3x+5y-z=-7 2x+3y+5z=21 3x+5y-1z=-7
x+y+z=-1 x-y-5z=-2 x+y+z=-1
2x+10z=6 2x+y-z=11 2x+11z=7

[4 mark]

Question Two

Matrix inverses can provide a simple and effective procedure for encoding and decoding messages. The numbers 1 – 26 have been assigned to letters of the alphabet as shown below. Additionally, the number 0 has been assigned to a blank to provide space between words.

Blank A B C D E F G H I J K L M
0 1 2 3 4 5 6 7 8 9 10 11 12 13

N O P Q R S T U V W X Y Z
14 15 16 17 18 19 20 21 22 23 24 25 26

The following encoding matrix has been used to encode a message:

A=[¦(2&3@1&2)]

Let B represent a (2 x 5) matrix containing the original message. The coded message is therefore derived by multiplying A and B, i.e. coded message = AB. The coded message is as follows:
AB=[¦(82&84@53&51)¦(33&53@21&29)¦(8@4)] Multiply the coded message by the inverse of the encoding matrix to find the original message.

[4 mark] What is the original (decoded) message?
[2 marks]

Due to a breach in security, QA headquarters has tasked you to create a new encoding matrix A. They require you to use a determinant of ?=-1 and the final inverse to be:
A^(-1)=[¦(-2&3@3&-4)] Calculate the new encoding matrix A.

[2 marks] What are the consequences to QA headquarters if the new encoding matrix is found to have a determinant of ?=0 ? What will you recommend?
[2 marks]

Question Three

Suppose an economy has two industries (agriculture and energy). These two industries depend on each other. Production of a dollar’s worth of agriculture requires an input of $0.60 from the agriculture sector and $0.30 from the energy sector. Production of a dollar’s worth of energy requires an input of $0.20 from the agriculture sector and $0.60 from the energy sector. Find the output of each sector that is needed to satisfy a final demand of $157 million for agriculture and $534 million for energy.

Let x represent the total output from agriculture
y represent the total output from energy

Write two equations to represent the internal and external demands for agriculture (x) and energy (y):

[1 mark] Find the technology matrix, M:

[1 mark]

Your technology matrix should have elements between 0 and 1. Explain why it is not possible to have elements in this matrix that are negative or greater than 1.
[2 marks]

Provide the proof that X = (I – M)-1D

[3 marks]

How much agriculture and energy does the economy need to produce in order to satisfy both internal and external demand?
[3 marks] Formula Sheet
Final production – input output analysis: X = (I – M)-1D
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