Exam 2: Systems of Linear Equations and Matrices

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. ​ ​ __________ (yes/no) ​ Find the solution or solutions to the system, if they exist. ​ __________ ​ __________ ​ __________ ​ __________

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Find the transpose of the matrix.

Solve the system of linear equations using the Gauss-Jordan elimination method. ​ ​

Let ​ Find . Please enter the elements of as decimals rounded to two decimal places.

Formulate but do not solve the problem. The annual returns on Sid Carrington's three investments amounted to $28,000: 6% on a saving account, 10% on mutual funds and 16% on bonds. The amount of Sid's investment in bouds was twice the amount of his investment in the savings account and the interest earned from his investment in bounds was equal to the dividends he received from his investment in mutual funds. Find how much money he placed in each type of investment. Let x be the money invested in saving account, y be the money investedr in mutual funds and z be the money invested in bonds.

Solve the linear system of equations ​ ​

Identify the column matrix. What is its transpose?

Solve the system of linear equations using the Gauss-Jordan elimination method.

A simple economy consists of three sectors: agriculture (A), manufacturing (M), and transportation (T).The input-output matrix for this economy is given by A M T Find the gross output of goods needed to satisfy a consumer demand for $100 million worth of agricultural products, $200 million worth of manufactured products, and $30 million worth of transportation.

Formulate but do not solve the problem. ​ The Johnson Farm has 600 acres of land allotted for cultivating corn and wheat. The cost of cultivating corn and wheat (including seeds and labor) is $46 and $23/acre, respectively. Jacob Johnson has $18,400 available for cultivating these crops. If he wishes to use all the allotted land and his entire budget for cultivating these two crops, how many acres of each crop should he plant?

A simple economy consists of three sectors: agriculture (A), manufacturing (M), and transportation (T). The input-output matrix for this economy is given by ​ A M T If the units are measured in millions of dollars, determine the amount of agricultural products consumed in the production of $300 million worth of manufactured goods.

Indicate whether the matrix is in row-reduced form.

Solve the matrix equation for the given matrices A, D and X. ​ ​ Round the elements of X to two decimal places. ​

Determine the value of k so that the following system of linear equations has a solution and then find the solution. ​ ​

Formulate but do not solve the problem. ​ The total number of passengers riding a certain city bus during the morning shift is 1,000. If the child's fare is $0.65, the adult fare is $0.35, and the total revenue from the fares in the morning shift is $440, how many children and how many adults rode the bus during the morning shift?

Write the system of linear equations in matrix form. - = - =

A dietitian wishes to plan a meal around three foods. The percent of the daily requirements of proteins, carbohydrates, and iron contained in each ounce of the three foods is summarized in the table. Determine how many ounces of each food the dietitian should include in the meal to meet exactly the daily requirement of proteins, carbohydrates, and iron (100% of each). ​

A dietitian plans a meal around three foods. The number of units of vitamin A, vitamin C, and calcium in each ounce of these foods is represented by the matrix M, where The matrices A and B represent the amount of each food (in ounces) consumed by a girl at two different meals, where Calculate .

A simple economy consists of two industries: agriculture and manufacturing. The production of 1 unit of agricultural products requires the consumption of 0.2 unit of agricultural products and 0.2 unit of manufactured goods. The production of 1 unit of manufactured products requires the consumption of 0.3 unit of agricultural products and 0.2 unit of manufactured goods. ​ Find the gross output of goods needed to satisfy a consumer demand for $50 million worth of agricultural products and $250 million worth of manufactured products. Round answers to two decimal places, if necessary. ​ $__________ million worth of agricultural goods ​ $__________ million worth of manufactured products.