1. Perform Basic Matrix Operations Chapter 3.5

2. History The problem below is from a Chinese book on mathematics written over 2000 years ago: There are three types of corn, of which three bundles of the first, two of the second, and one of the third make 39 measures. Two of the first, three of the second and one of the third make 34 measures. And one of the first, two of the second and three of the third make 26 measures. How many measures of corn are contained of one bundle of each type?

3. History

4. The use of variables to represent numbers did not come into wide use until about 300 years ago, so this was problem was not represented as we have written it Instead, the author of the text set up a table like the one below 123 232 311 263439

5. History Next, the author performs what today we would call row operations (though he uses columns) on the middle column to obtain the table below The effect is to have eliminated the x-term in the middle equation 103 252 321 262439

6. History Continuing in this manner, he is able to find the solution to the problem The point to notice is that he did this in a table and without the use of variables It was not until 1850 that this arrangement (changed somewhat, as you will see) came to be called a matrix

7. History

9. Matrices A few weeks back, we found that we needed a way to make certain that a distance measure on the number line is always a positive number To do this, we invented the idea of an absolute value and came up with a symbol to indicate when we wanted to take an absolute value However, we were later able to study the absolute value apart from the idea of distance on a number line The same will be true of matrices They came about as a way to solve systems of equations But we will be able to study them without regard to a system of equations

10. Matrices What mathematicians (in the 19 th century and afterwards) found interesting about matrices is that they behave a lot like numbers. We can perform operations on them, like adding, subtracting, and multiplying These operations obey many of the number properties (though not all for every operation, and some properties are special to matrices) In this section you will learn about two basic operations on matrices and the properties of these operations

11. The Basics

12. 3 rows 4 columns

13. The Basics 2 nd row 4 th column

14. The Basics

16. Adding & Subtracting Since matrices are a new kind of mathematical object, we must define what we mean by addition and subtraction We add or subtract matrices by adding or subtracting the elements in corresponding positions, with the results recorded in the corresponding positions This means that we may only add and subtract matrices of the same dimensions and the result is a matrix of the same dimension

17. Adding & Subtracting

20. Scalar Multiplication We will define two kinds of matrix multiplication Multiplication of a matrix by another matrix is called matrix multiplication, and you will learn about this in the next section Multiplication of a matrix by a real number is called scalar multiplication A scalar is just a regular number (not a matrix) Scalar multiplication is performed by multiplying each element in the matrix by the scalar (much like using the distributive property)

21. Scalar Multiplication

23. Guided Practice

27. Matrix Properties

28. Matrices also have additive inverses as well as a zero matrix The zero matrices are different depending on the dimensions, but they all have a zero in every position Having an additive inverse means that we can set up and solve a matrix equation

29. Solve a Matrix Equation

33. Guided Practice

37. Exercise 3.5 Page 191, #5-27 odds, Page 193, #7,8 (14 total problems)