Exam 4: Matrices

A closed model for an economy has a manufacturing industry, utilities industry, and households industry. Each unit of manufacturing output uses 0.6 unit of manufacturing input, 0.4 unit of utilities input, and 0.2 unit of households input. Each unit of utilities output requires 0.3 unit of manufacturing input, 0.5 unit of utilities input, and 0.2 unit of households input. Each unit of household output requires 0.2 unit each of manufacturing and utilities input and 0.4 unit of households input. Write the augmented matrix for this closed model of the economy. ​

Card tables are made by joining 4 legs and a top using 4 bolts. The legs are each made from a steel rod. The top has a frame made from 4 steel rods. A cover and four special clamps that brace the top and hold the legs are joined to the frame using a total of 8 bolts. The parts-listing matrix for the card table assembly is given by ​ ​ If an order is received for 10 card tables, 4 legs, 1 top, 1 cover, 7 clamps, and 12 bolts, how many of each primary assembly item are required to fill the order? ​

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

If and , are C and D inverse matrices?

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 12% for group I, 5% 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 ​ ​ ​ Find the matrix that gives the new weights and measures at the end of this period by computing , where is the 3×3 identity matrix. ​

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) ​ ​ Which industry is most dependent on its own goods for its operations? ​

Use matrix F to find F², if possible. ​ ​

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

A primitive economy has a mining industry and a fishing industry, with technology matrix A. ​ ​ ​ If surpluses of 132 units of mining output and 27 units of fishing output are desired, find the gross production of each industry. ​

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

Use technology to find the determinant. ​ ​

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 0.3 unit of government input, 0.2 unit of profit sector input, 0.2 unit of nonprofit sector input, and 0.4 unit of households input. Each unit of profit sector output requires 0.2 unit of government input, 0.3 unit of profit sector input, 0.2 unit of nonprofit sector input, and 0.4 unit of households input. Each unit of nonprofit sector output requires 0.2 unit of government input, 0.2 unit of profit sector input, 0.2 unit of nonprofit sector input, and 0.6 unit of households input. Each unit of households output requires 0.05 unit of government input, 0.2 unit of profit sector input, 0.2 unit of nonprofit sector input, and 0.25 unit of households input. Write the augmented matrix for this closed model of the economy. ​

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. ​ ​Males Females ​​ ​

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. ​

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) ​ ​ Which industry is least dependent on its own goods? ​

What is the sum of matrix M and its negative? ​ ​

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

Is it true for matrices (as it is for real numbers) that multiplication by a zero matrix only sometimes gives is a result of a zero matrix provided the multiplication is defined?

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. ​ , ,

Multiply the matrices. ​ ​