Mathematical Techniques - Matrix Algebra

​​Explain the following terms as used in the input-output analysis: 

​​Using appropriate illustrations, explain the following terms as used in matrices: 

​ ​ ​​The 2 x 2 matrices; A, X and B are given as follows:

​ ​ ​ ​ ​ ​ ​ ​​An economy produces food, clothing and shelter. To produce one unit of food, it requires 0.6 units from food sector, 0.2 units from clothing sector and 0.1 units from the shelter sector. To produce one unit of clothing, it requires 0.1 units from the food sector, 0.4 units from clothing sector and 0.2 units from the shelter sector. To produce one unit of shelter it requires 0.1 units from the food sector, 0.4 units from the clothing sector and 0.4 units from the shelter sector. 

​​Highlight FOUR applications of Markov chain analysis in business and finance.

​​A trader purchased 7 cartons of apples and 4 cartons of bananas at a total cost of Sh.32,500 in the month of January.In the month of February, he purchased 9 cartons of apples and 6 cartons of bananas at a total cost of Sh.43,500.

Required:

Calculate the cost of a carton of apples and a carton of bananas using matrix algebra

​ ​ ​ ​ ​ ​​Using the information given below, determine the level of output necessary from each sector to support a final demand of 500 units for sector X1, 1,600 units for sector X2 and 2,000 units for sector X3:

​ ​ ​​The banking industry in a given country is controlled by three banks Faida Bank, Akiba Bank and Ahadi Bank. As at 31 December 2020, Faida Bank controlled 30% of the market share, Akiba Bank controlled 50% of the market share and Ahadi Bank controlled 20% of the market share. 

​ ​​A small economy has two sectors, X₁ and X2 producing a single product for their internal and external demand (in units), as summarised in the following transaction matrix.

​ ​ ​ ​​A chemical reaction in a processing plant is given by: 

​​A salesman earns a fixed monthly basic salary and a commission that is directly proportional to the number of units sold in the month. During the months of February 2019 and March 2019, the salesman's total earnings were Sh.60,000 and Sh.70,000 respectively. The number of units sold by the salesman in the months of February 2019 and March 2019 were 100 and 250 respectively. During the month of April 2019, the salesman sold 400 units. 

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A certain firm has three main departments namely; steel, motor vehicles and construction. The three departments are interdependent. Each unit of output from the steel department requires 0.2, 0.3 and 0.4 units from steel, motor vehicles and construction departments respectively. Each unit of output from motor vehicles department requires 0.2, 0.4 and 0.2 units from steel, motor vehicles and construction departments respectively. A unit of output from the construction department requires 0.3, 0.4 and 0.1 units from steel, motor vehicles and construction departments respectively. The final demand of the firm comprises 20 million, 50 million and 30 million units of output from steel, motor vehicles and construction departments respectively.

​ ​​Excellent Products Limited manufactures four products, A, B, C and D, using four machines, M₁, M₂, M, and M;. The total outputs of the four products from the four machines are as shown below:

​ ​ ​ ​​Soda Baridi Limited has a computerised customer billing system. Customers' accounts are classified as being paid, delinquent or bad debt. The company has a total of 1,500,000 customer accounts. A computer program was developed to display transition of accounts among the three categories. The output from the program is summarised below:

​​A certain audit firm has two categories of employees, auditors and assistant auditors. The total monthly salary of I auditor and 5 assistant auditors amount to Sh.456.755 whereas the total monthly salary of 3 auditors and 9 assistant auditors amount to Sh.985,005. The firm has a total of 6 auditors and 14 assistant auditors. The employees contribute 12 per cent of their monthly salaries towards their sacco society.

Required:
(i) The monthly salary of an auditor and an assistant auditor, using matrix algebra,
(ii) The employees' total monthly contribution towards their sacco society.

Star Manufacturers Limited specialises in the production of two products, A and B. The manufacturer sells the products at a fixed selling price to its customers. The following table shows the requirements for production of products A and B:


Product A is sold for Sh.2.080 per unit whereas product B is sold for Sh.7.939 per unit. The variable costs of production are uncertain with the following margins of error:


Required:
Using matrix algebra, determine:

(i) The total expected revenue.
(ii) The expected maximum profit.
(iii) The expected minimum profit.

​ ​ ​ ​ ​ ​​The research industry in your country has three market research firms namely X, Y and Z which provide research services. The following data has been collected in relation to the flow of clients among the three firms: