Solving Systems of Linear Equations using Matrix Methods

Matrices have a number of interesting mathematical attributes, such as their dimensions, how they can be derived from linear systems, and the kinds of operations that can be performed on them.

1. Matrix methods can be used to solve linear programming problems. A linear programming problem is used to find an optimal solution, subject to stated restraints.

One typical application is to maximize profits. For example, a beauty parlor provides both highlighting and permanent wave services. It costs $5 in materials and requires 30 minutes to provide highlighting. However, it costs $12 in materials but requires 80 minutes to provide a perm. The store has at most $120 in materials and 800 minutes in labor per day to expend.

How many highlighting services and how many perms can the beauty parlor perform daily?

2. You are given the following system of linear equations:
3x - 2y + z = 2
-x + y = 3
-2y + 6 = -1

a. Provide a coefficient matrix corresponding to the system of linear equations.
b. What is the inverse of this matrix?
c. What is the transpose of this matrix?
d. Find the determinant for this matrix.

3. Calculate the following for
a. A * B
b. -3A
c. A-1

4. Solve the following linear system using Gaussian elimination.
Show work.
2y + z = 4
x+ y +z = 6
2x + y + z = 7

5. Solve the following linear system for x using Cramer's rule.
Show work.
4x - y + z = -5
2x + 2y + 3z = 10
5x - 2y + 6z = 1

Human populations can be modeled using an exponential growth function. Use the Library for research or other resources to find the following:
- current world population
- current U.S. population
- current growth rate of the world population
- current growth rate of the U.S. population

Starting in the current year, determine the following:

1. Based on current growth rates, how long will it take world population to reach exactly 7 billion? How long to reach exactly 10 billion?

2. Based on current growth rates, how long will it take for the U.S. population to reach exactly 300 million? How long for exactly 400 million?