Exam 4: Matrices

A manufacturer of table saws has three models: Deluxe, Premium, and Ultimate, which must be painted, assembled, and packaged for shipping. The table gives the number of hours required for each of these operations for each type of table saw. If the manufacturer has 98 hours available per day for painting, 150 hours for assembly, and 36 hours for packaging, how many of each type of saw can be produced each day? ​ ​

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

Write the transpose of matrix F. ​ ​

Use inverse matrices to solve the system of linear equations. ​ ​

Use determinants to decide whether the matrix has an inverse. ​

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

Suppose products A and B are made from plastic, steel, and glass, with the number of units of each raw material required for each product given by the table below. ​ ​ Because of transportation costs to the firm's two plants, X and Y, the unit costs for some of the raw materials are different. The table below gives the unit costs for each of the raw materials at the two plants. ​ Using the information just given, find the total cost of producing each of the products at each of the factories. ​​

What is the order of matrices A, B, C, D, E? ​ , , , , ​

For , can ? Explain. ​

Suppose that an economy can be represented by the technology matrix below. If surpluses of 180 units of fishing output, 93 units of agricultural goods, and 40 units of mining goods are desired, find the gross production to the nearest unit of each industry. ​ ​

The following tables give the rank and number of registered dogs for the top ten breeds for 1995 and their 2002 data. Which breed had the largest increase in the number of registrations? ​ 1995 2002

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? ​

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

Let matrix A represent the sales (in thousands of dollars) for the Walbash Company in 2004 in various cities, and let matrix B represent the sales (in thousands of dollars) for the same company in 2005 in the same cities. Write the matrix that represents the total sales by type (wholesale and retail) and city (Chicago, Atlanta, and Memphis) for both years. ​ ; ​

We have encoded messages by assigning the letters of the alphabet the numbers 1-26, with a blank assigned the number 27, and by using an encoding matrix A that converts the numbers to the coded message. To decode any message encoded by A, we must find the inverse of A, denoted by A^-1, and multiply A^-1 times the coded message. Use this code and the given encoding matrix A to find the matrix that can be used to decode the given message. ​ The code matrix is . The message is 33, 132, 214, 22, 110, 203, 34, 132, 201.

A closed model for an economy identifies government, the profit sector, the nonprofit sector, and households as its industries. Each unit of government output requires unit of government input, unit of profit sector input, unit of nonprofit sector input, and unit of households input. Each unit of profit sector output requires unit of government input, unit of profit sector input, unit of nonprofit sector input, and unit of households input. Each unit of nonprofit sector output requires unit of government input, unit of profit sector input, unit of nonprofit sector input, and unit of households input. Each unit of households output requires unit of government input, unit of profit sector input, unit of nonprofit sector input, and unit of households input. Find the gross production for each industry. ​

A psychologist studying the effects of nutrition on the behavior of laboratory rats is feeding one group a combination of three foods: I, II, and III. Each of these foods contains three additives, A, B, and C, that are being used in the study. Each additive is a certain percentage of each of the foods as follows: ​ Foods ​ If the diet requires 53 g per day of A, g per day of B, and g per day of C, find the number of grams per day of each food that must be used.​

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

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

A system of equations may have a unique solution, an infinite number of solutions, or no solution. Use matrices to find the general solution of the following system, if a solution exists.