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.

Copy the questions to a Microsoft Word document and use an equation editor to enter the answers. Please answer the following questions about matrices.

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 $90 in materials and 580 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

1. Provide a coefficient matrix corresponding to the system of linear equations.

2. What is the inverse of this matrix?

3. What is the transpose of this matrix?

4. Find the determinant for this matrix.

3. Calculate the following for

1. A * B
2. -3A
3. 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