Exam 4: Matrices

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) ​ ​ How many units of fuels were required to produce 1,000 units of manufactured goods? ​

A man has $282,000 invested in three properties. One earns 12%, one 10%, and one 8%. His annual income from the properties is $27,000 and the amount invested at 8% is twice that invested at 12%. How much is invested in each property?

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

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. ​ ​

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. Suppose ALCO decides to pay 80% of all debts due in 30 days and to increase the debts due in 60 days by 20%. Write a matrix that gives the debts after these transactions are made. ​ ​ ​

Write the matrix that is the negative of matrix B. ​ ​

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

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

A trust account manager has $220,000 to be invested. The investment choices have current yields of 8%, 7%, and 10%. Suppose that the investment goal is to earn interest of $16,300, and risk factors make it prudent to invest some money in all three investments. Let x denote the amount invested at 8 %, y denote the amount invested at 7%, and z denote the amount invested at 10%. Find a general description for the amounts invested at the three rates. ​

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

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. ​ , ,

For the economy, what industry is most dependent on utilities? ​ ​

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) ​ ​ How many units of fuels were required to produce 1,000 units of manufactured goods? ​

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

​In order to rank the five members of its chess team for play against another school, the coach draws the following diagram. An arrow from 1 to 2 means player 1 has defeated player 2. ​​ Let matrix A represent the matrix with the following elements: . Let matrix B represent the matrix with the following elements: . The player i is the top-ranked player if the sum of row i in the matrix A + B is the largest. What is the number of this player? ​

Find the inverse matrix, if one exists.

One sector of an economy consists of a mining industry and a manufacturing industry, with technology matrix A. ​ ​ Surpluses of 32 units of mining output and 263 units of manufacturing output are desired. Find the gross production of each industry. ​

Suppose the weights (in grams) and lengths (in centimeters) of three groups of laboratory animals are given by matrix A, where column 1 gives the lengths and each row corresponds to one group. ​ ​ ​If the increase in both weight and length over the next two weeks is 19% for group I, 9% for group II, and 3% for group III, then the increases in the measures during the 2 weeks can be found by computing GA, where ​ What are the increases in respective weights and measures at the end of these 2 weeks? ​

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. ​ , ,

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