Exam 2: Systems of Linear Equations and Matrices

Solve for u, x, y, and z in the matrix equation.

William and Michael's stock holdings are given by the matrix At the close of trading on a certain day, the prices (in dollars per share) of the stocks are given by the matrix .

Identify the row matrix. What is its transpose? Row matrix: __________ (A,B,C or D) Transpose: __________

A simple economy consists of two industries: agriculture and manufacturing. The production of 1 unit of agricultural products requires the consumption of 0.3 unit of agricultural products and 0.2 unit of manufactured goods. The production of 1 unit of manufactured products requires the consumption of 0.4 unit of agricultural products and 0.1 unit of manufactured goods. ​ Find the value of the goods consumed in the internal process of production needed to satisfy a consumer demand for $50 million worth of agricultural products and $300 million worth of manufactured products. Round answers to two decimal places. ​

Compute the product. ​ ​

Identify the square matrix. What is its transpose? Square matrix: __________ (A,B,C or D) Transpose: __________

Compute the product. ​ ​

In this problem matrix A is an input-output matrix associated with an economy, and matrix D (units in millions of dollars) is a demand vector. Find the final outputs of each industry so that the demands of both industry and the open sector are met. The final output of first industry is $ __________ million. The final output of second industry is $ __________ million.

Cindy regularly makes long distance phone calls to three foreign cities: London, Tokyo, and Hong Kong. The matrices A and B give the lengths (in minutes) of her calls during peak and nonpeak hours, respectively, to each of these three cities during the month of June. and The costs for the calls (in dollars per minute) for the peak and nonpeak periods in the month in question are given, respectively, by the matrices Compute the matrix AC + BD

Pivot the system about the bolded element. ​ ​

A dietitian wishes to plan a meal around three foods. The meal is to include 6,600 units of vitamin A, 3,980 units of vitamin C, and 200 units of calcium. The number of units of the vitamins and calcium in each ounce of the foods is summarized in the accompanying table: Determine the amount of each food the dietitian should include in the meal in order to meet the vitamin and calcium requirements.

Consider a simple economy consisting of three sectors: food, clothing, and shelter. The production of 1 unit of food requires the consumption of 0.1 unit of food, 0.4 unit of clothing, and 0.4 unit of shelter. The production of 1 unit of clothing requires the consumption of 0.4 unit of food, 0.2 unit of clothing, and 0.3 unit of shelter. The production of 1 unit of shelter requires the consumption of 0.2 unit of food, 0.1 unit of clothing, and 0.1 unit of shelter. Find the level of production for each sector in order to satisfy the demand for $300 million worth of food, $70 million worth of clothing, and $270 million worth of shelter. Round the answers to the nearest tenth. $ __________ million worth of food $ __________ million worth of clothing $ __________ million worth of shelter

Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations, determine whether the system has a solution and find the solution or solutions to the system, if they exist.

A private investment club has $200,000 earmarked for investment in stocks. To arrive at an acceptable overall level of risk, the stocks that management is considering have been classified into three categories: high-risk, medium-risk, and low-risk. Management estimates that high-risk stocks will have a rate of return of 15% per year; medium-risk stocks, 10% per year; and low-risk stocks, 4% per year. The members have decided that the investment in low-risk stocks should be equal to the sum of the investments in the stocks of the other two categories. Determine how much the club should invest in each type of stock if the investment goal is to have a return of $17,000/year on the total investment. (Assume that all the money available for investment is invested.) High-risk investment $__________ Medium-risk investment $__________ Low-risk investment $__________

A dietitian wishes to plan a meal around three foods. The meal is to include 9,600 units of vitamin A, 2,460 units of vitamin C, and 1,080 units of calcium. The number of units of the vitamins and calcium in each ounce of the foods is summarized in the accompanying table. Can such a meal be planned around the foods? 60

Rainbow Harbor Cruises charges $8/adult and $4/child for a round-trip ticket. The records show that, on a certain weekend, 900 people took the cruise on Saturday and 1,300 people took the cruise on Sunday. The total receipts for Saturday were $6,000, and the total receipts for Sunday were $8,400. Determine how many adults and children took the cruise on Saturday and on Sunday. ​

The relationship governing the intraindustrial and interindustrial sales and purchases of four basic industries - agriculture (A), manufacturing (M), transportation (T), and energy (E) - of a certain economy is given by the following input-output matrix. A M T E How many units of energy are required to produce 5 units of all goods in the economy?

Mr. and Mrs. Garcia have a total of $300,000 to be invested in stocks, bonds, and a money market account. The stocks have a rate of return of 10%/year, while the bonds and the money market account pay 6% and 2%/year, respectively. They have stipulated that the amount invested in stocks should be equal to the sum of the amount invested in bonds and 3 times the amount invested in the money market account. How should the Garcias allocate their resources if they require an annual income of $24,000 from their investments?

Solve the system of linear equations using the Gauss-Jordan elimination method. If there is no solution, answer none. ​

Determine the value of k so that the following system of linear equations has infinitely many solutions and then find the solutions: ​