Exam 4: Matrices

Use the following matrices to perform the indicated matrix operations, if possible. ​ ​ ; ​

In the analysis of traffic flow, a certain city estimates the following situation for the "square" of its downtown district. In the following figure, the arrows indicate the flow of traffic. If x₁ represents the number of cars traveling between intersections A and B, x₂ represents the number of cars traveling between B and C, x₃ the number between C and D, and x₄ the number between D and A, we can formulate equations based on the principle that the number of vehicles entering an intersection equals the number leaving it. That is, for intersection A we obtain ​ Formulate equations for the traffic at B, C, and D. Solve the system of these four equations. ​ ​

If and , does ?

What is element ? ​ ​

The given matrix is an augmented matrix representing a system of linear equations. Find the solution of the system. ​ ​

Set up the system of equations and then solve it by using inverse matrices. Transportation Ace Trucking Company has an order for three products, A, B, and C to be delivered. The table gives the volume in cubic feet, the weight in pounds, and the value for insurance in dollars for a unit of each of the products. If the carrier can carry 8,000 cu ft and 12,400 lb and is insured for $55,800, how many units of each product can be carried? ​

A clothing manufacturer has factories in Atlanta, Chicago, and New York. Sales (in thousands) during the first quarter are summarized in the matrix below. ​ ​ During this period, the selling price of a coat was $100, of a shirt $35, of a pair of pants $50, and of a tie $25. Use matrix multiplication to find the total revenue received by each factory. ​

The sales department of an auto dealership charges 10% of its total monthly costs to the service department, and the service department charges 20% of its total monthly costs to the sales department. During a given month, the direct costs are $87,100 for sales and $49,600 for service. Find the total costs of each department. ​

Use the matrices below. Perform the indicated operations. ​ ​ ​ ​

​When a firm buys another company, the company frequently has some outstanding debt that the purchaser must pay. Consider three companies, A, B, and C, that are purchased by Maxx Industries. The following table gives the amount of each company's debt, classified by the number of days remaining until the debt must be paid. Suppose Maxx Industries pays 20% of the amount owed on each loan. Write the matrix that would give the remaining debt in each category. ​

The following technology matrix describes the relationship of certain industries within the economy to each other. (A&F, agriculture and food; RM, raw materials; M, manufacturing; F, fuels industry; U, utilities; SI, service industries) ​ ​ Which is most dependent on the service industry? ​

Describe the procedure for finding the general solution of a system of three linear equations involving p, q, and r if its reduced matrix indicate has one row of zeros at the bottom. You may assume that the coefficients in the augmented matrix correspond to variables in alphabetical order from left to right. ​

Use technology to solve the system of equations, if a solution exists. ​

Use matrix A to find A³, if possible. ​ ​

Suppose a primitive economy consists of two industries, farm products and farm machinery. Suppose also that its technology matrix is ​ If surpluses of 97 units of farm products and 18 units of farm machinery are desired, find the gross production of each industry. ​

The tables below give the numbers of some species of threatened and endangered wildlife in the United States and in foreign countries in 2003. Let A represent the matrix that contains the number of each of these species in the United States in 2003 and B represent the matrix that contains the number of each of these species outside the United States in 2003. Find the matrix B - A. What do the negative entries in matrix B - A mean? ​ ​United States ​Foreign ​ ​

The following technology matrix describes the relationship of certain industries within the economy to each other. (A&F, agriculture and food; RM, raw materials; M, manufacturing; F, fuels industry; U, utilities; SI, service industries) ​ ​ Which industry would be most affected by a rise in the cost of utilites? ​

Use row operations on the augmented matrix to solve the given system of linear equations. ​ ​

Suppose that a government study showed that 80% of urban families remained in an urban area in the next generation (and 20% moved to a rural area), whereas 70% of rural families remained in a rural area in the next generation (and 30% moved to an urban area). This means that if represents the current percents of urban families u₀ and of rural families r_0, then represents the percents of urban families u and rural families r one generation from now. Currently the population is 70% urban and 30% rural. Find the percents in each group one generation before this one. ​

The given matrix is an augmented matrix representing a system of linear equations. Find the solution of the system. ​ ​