Exam 3: Matrices

A closed model for an economy has a manufacturing industry, utilities industry, and households industry. Each unit of manufacturing output uses 0.5 unit of manufacturing input, 0.4 unit of utilities input, and 0.1 unit of households input. Each unit of utilities output requires 0.4 unit of manufacturing input, 0.5 unit of utilities input, and 0.1 unit of households input. Each unit of household output requires 0.4 unit each of manufacturing and utilities input and 0.2 unit of households input. Find the gross production for each industry. ​

The following technology matrix for a simple economy describes the relationship of certain industries to each other in the production of 1 unit of product. ​ ​ How many units of utilities are required to produce 30 units of agricultural products? ​

An irrigation system allows water to flow in the pattern shown in the figure below. Water flows into the system at A and exits at B, C, D, and E with the amounts shown. Using the fact that at each point the water entering equals the water leaving, formulate an equation for water flow at each of the five points and solve the system. ​ ​

The tables below give the numbers of some species of threatened and endangered wildlife in the United States and in foreign countries in 2003. Find a matrix with the total number of these species. Assume that U.S. and foreign species are different. ​

Use technology to find the determinant. ​ ​

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 is least dependent on its own goods? ​

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

Is it true for matrices (as it is for real numbers) that multiplication by a zero matrix only sometimes gives is a result of a zero matrix provided the multiplication is defined?

Suppose a simple economy with only an agricultural industry and a steel industry has the following technology matrix. ​ ​ If surpluses of 15 units of agricultural products and 34 units of steel are desired, find the gross production to the nearest unit of each industry. ​

Each ounce of substance A supplies of the required nutrition a patient needs. Substance B supplies of the required nutrition per ounce and substance C supplies of required nutrition per ounce. If digestive restrictions require that substances A and C be given in equal amounts and that the amount of substance B be one-fifth of these other amounts, find the number of ounces of each substance that should be in the meal to provide 100% nutrition. ​

The given matrix is an augmented matrix used in the solution of a system of linear equations. What is the solution of the system?

A car dealer can buy midsize cars for 11% under the list price, and he can buy luxury cars for 14% under the list price. The following table gives the list prices for two midsize and two luxury cars. ​ Write these data in a matrix and multiply it on the left by the matrix ​ ​ ​

The given matrix is an augmented matrix used in the solution of a system of linear equations. What is the solution of the system?

Suppose that an economy has the technology matrix below. If surpluses of 56 units of manufactured goods and 104 units of agricultural goods are desired, find the gross production of each industry. ​ ​

Substitute the given values of x, y, and z into the matrix equation and use matrix multiplication to determine if the values satisfy the equation. ​ , ,

If and , does ?

Use technology to find the product AB of the following matrices. ​ ​

Use the matrices to find BA, if possible. ​ ​ ​

Suppose the shipping department of a firm charges 20% of its total monthly costs to the printing department and that the printing department charges 10% of its total monthly costs to the shipping department. If the direct costs of the shipping department are $18,000 and the direct costs of the printing department are $11,000, find the total costs for each department. Round your answer to the nearest hundredths. ​

Set up the system of equations and then solve it by using inverse matrices. ​ Investment A company offers three mutual fund plans for its employees. Plan I consists of 4 blocks of common stock and 2 municipal bonds. Plan II consists of 8 blocks of common stock, 4 municipal bonds, and 6 blocks of preferred stock. Plan III consists of 14 blocks of common stock, 6 municipal bonds, and 6 blocks of preferred stock. An employee wants to combine these plans so that she has 82 blocks of common stock, 38 municipal bonds, and 36 blocks of preferred stock. How many units of each plan does she need? ​