1. Systems and Matrices Chapter 5 College Algebra Ocampo

2. Introduction To make predictions about the future, professionals in many fields attempt to determine relationships between different factors. It is often desirable or even necessary to use more than two variables to model a situation in a field such as business, science, psychology, engineering, education, and sociology, to name a few. When this is the case, we write and solve a system of equations in order to answer questions about the situation.

3. Consider This… You are a tax auditor for the Internal Revenue Service. Your boss tells you to investigate the taxes filed regarding some investments made by the company, Charles Schwab. The information for you to investigate is as follows:

4. The Problem Charles Schwab invested a total of $400,000 in three different places. The company claims that they invested $380,000 in an insurance bond that gained an annual interest of 1.5%, $10,000 in a fortune 500 company that gained an annual interest of 2%, and $10,000 in a Chase savings account that gained 2.5% interest annually. After the first year, the company filed that their total interest gain was $8,250, claiming that $5,700 of that interest is non-taxable since it is categorized under insurance.

5. This problem involves three variables and three separate equations, which is more difficult to solve than 2-variable scenarios you may have been used to in your study of mathematics thus far. Wouldn’t it be great if there was a way to solve this in a quick and simple way?

6. In This Chapter, You Will Learn… The properties of matrices (5.7) Matrix inverses (5.8) How to perform operations on matrices (5.1) How to solve systems of equations with matrices (5.2) How to determine what types of solutions a matrix will have (5.3)

7. Properties of Matrices Section 5.7

8. What is a Matrix? A matrix is an ordered set of numbers listed rectangular form Usually named with a capital letter, such as A, B, or C Example. Let A denote the matrix [2 5 7 8] [5 6 8 9] [3 9 0 1] This matrix A has three rows and four columns. We say it is a 3x4 matrix. We denote the element (entry) on the second row and fourth column with a 2,4

9. Types of Matrices Square matrix If a matrix A has n rows and n columns then we say it's a square matrix. Row matrix - A matrix with one row Column matrix - A matrix with one column Matrices of the same kind Matrix A and B are of the same kind if and only if A has as many rows as B and A has as many columns as B

10. Equality of Matrices Two matrices are equal if… 1. They are the same size 2. Corresponding (matching in position) elements are equal Example: Below, 1 = 3 but 1 ≠ 2

11. Are These Matrices Equal? Matrix AMatrix B No! a 2,3 is not equal to b 2,3

12. If These are Equal… What is the value of x? What is the value of y? What is the value of z? Matrix A=Matrix B X = 12 100 = 20y y = 5 3z = 4 Z = 4/3

13. Addition/Subtraction of Matrices Simply add/subtract corresponding numbers. You can only add or subtract matrices if they are same type matrices. Example: ABCABC You can add A and B together, but you cannot add C to anything here.

14. Example Problems Perform the following operations. A + B B – A A – B A – C ABC