Exam 3: Matrices

An economy has a manufacturing industry and a banking industry. Each unit of manufacturing output requires inputs of 0.5 unit of manufacturing and 0.2 unit of banking. Each unit of banking output requires inputs of 0.3 unit each of manufacturing and banking. If surpluses of 133 units of manufacturing and 100 units of banking are desired, find the gross production of each industry. ​

Suppose the technology matrix for a closed model of an economy is given by matrix A. Find the gross productions for the industries. ​ ​

Use the matrices below. Perform the indicated operations. ​ I·F·Z ​

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

Use inverse matrices to find the solution of the system of equations. ​ ​

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

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

The figure that follows depicts the mean (average) flow time for a job when the critical-ratio rule is used to dispatch workers at several machines required to complete the job. In the figure, WHN 1, WHN 2, and WHN 3 represent three different rules for determining when a worker should be transferred to another machine, and Where represents six rules for determining the machine to which the worker is transferred. Construct a 3×6 matrix A to represent these data, with entries rounded to the nearest integer. ​ ​

Find the inverse matrix, if one exists.

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

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 to find , if possible. ​ ​ ​

The following tables give the death rates, per 100,000 population, by age for selected years for males and females. If matrix M gives the male data and matrix F gives the female data, use matrix operations to find the death rate per 100,000 for all people in the age categories given and for the given years. ​ ​

A brokerage house offers three stock portfolios. Portfolio I consists of 2 blocks of common stock and 1 municipal bond. Portfolio II consists of 4 blocks of common stock, 2 municipal bonds, and 3 blocks of preferred stock. Portfolio III consists of 2 blocks of common stock, 1 municipal bond, and 3 blocks of preferred stock. A customer wants 16 blocks of common stock, 8 municipal bonds, and 6 blocks of preferred stock. Let x equal the number of units of Portfolio I, y equal the number of units of Portfolio II, and z equal the number of units of Portolio III. If the numbers of the three portfolios offered must be integers, find the general solution that can be used to determine all possible offerings. ​

The inverse of matrix A is given. Use the inverse to solve for X. ​ Solve . ​

For , can FF^T = F^TF? Explain. ​

Use matrix A to find A², if possible. ​ ​

Ace, Baker, and Champ are being purchased by ALCO, Inc., and their outstanding debts must be paid by the purchaser. The matrix below gives the amounts of debt and the terms for the companies being purchased. If ALCO pays 35% of the amount owed on each debt, write the matrix giving the remaining debts. ​ ​ ​

A system of linear equations and a reduced matrix for the system are given. Use the reduced matrix to find the general solution of the system, if one exists ​ ​​ ​​ ​

​Using the data in the following tables, Let matrix A represent imports (in billions of dollars) to U.S. from country groupings, and let matrix B represent exports (in billions of dollars) from U.S. to country groupings. Find the total U.S. trade with these groupings by finding A + B. ​