# Matrices, Inverse, Transpose, Determinant, Gaussian Elimination and Cramer's Rule

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.

1. For example, consider an accountant who prepares tax returns. Suppose a form 1040EZ requires $12 in computer resources to process and 22 minutes of the accountant's time. Assume a form 1040A takes $25 in computer resources and needs 48 minutes of the accountant's time. If the accountant can spend $630 on computer resources and has 1194 minutes available, how many forms of 1040EZ and 1040A can the accountant process?

2. You are given the following system of linear equations:

x - y + 2z = 13

2x + 2y - z = -6

-x + 3y + z = -7

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 2 -3 -4 1

b. -4A A= and B=

c. AT 7 4 6 5

4. Solve the following linear system using Gaussian elimination.

Show work.

3x + y - z = -5

-4x + y = 6

6x - 2y + 3z = 2

5. Solve the following linear system for x using Cramer's rule.

Show work.

x + 2y - 3z = -21

2x - 6y + 8z = 73

-x - 2y + 4z = 28

#### Solution Preview

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.

For example, consider an accountant who prepares tax ...

#### Solution Summary

Matrces, inverse, transpose, determinant, Gaussian elimination and Cramer's rule are investigated in this solution, which is given in a Word document.