Exam 9: Systems of Equations and Inequalities

Let be the identity matrix of order 2, and let . Find the polynomial for the given matrix A. (In the study of matrices, is the characteristic polynomial of A, and the zeros of are the characteristic values (eigenvalues) of A.)

Find the determinant of the matrix.

Three solutions contain a certain acid. The first contains 10% acid, the second 30%, and the third 50%. A chemist wishes to use all three solutions to obtain a 10-liter mixture containing 24% acid. If the chemist wants to use twice as much of the 50% solution as of the 30% solution, how many liters of each solution should be used?

Use Cramer's rule, whenever possible, to solve the system.

A shop specializes in preparing blends of gourmet coffees. From Colombian, Costa Rican, and Kenyan coffees, the owner wishes to prepare 3-pounds bags that will sell for $8.50. The cost per pound of these coffees is $10, $6, and $8, respectively. The amount of Colombian is to be three times the amount of Costa Rican. Find the amount of each type of coffee in the blend.

Find the inverse of the matrix if it exists.

Find the maximum and minimum values of the objective function on the region in the figure.

Use matrices to solve the system.

Find, if possible, .

Find the inverse of the matrix if it exists.

Use the method of substitution to solve the system.

Find the maximum and minimum values of the objective function on the region in the figure.

Three solutions contain a certain acid. The first contains 10% acid, the second 30%, and the third 50%. A chemist wishes to use all three solutions to obtain a 100-liter mixture containing 28% acid. If the chemist wants to use twice as much of the 50% solution as of the 30% solution, how many liters of each solution should be used?

Find the inverse of the matrix if it exists.

Use matrices to solve the system.

Use the method of substitution to solve the system.

Find the determinant of the matrix.

Sketch the graph of the inequality.

The data in the table are generated by the function . Approximate the unknown constants a and b to four decimal places. x 1 2 3 4 F ( x ) 0)71939 0.41687 0.24157 0.13998

Find the inverse of the matrix if it exists.