1. Chapter 8 Matrices and Determinants Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 8.3 Matrix Operations and Their Applications

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Use matrix notation. Understand what is meant by equal matrices. Add and subtract matrices. Perform scalar multiplication. Solve matrix equations. Multiply matrices. Model applied situations with matrix operations. Objectives:

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Notations for Matrices An array of numbers, arranged in rows and columns and placed in brackets, is called a matrix. We can represent a matrix in two different ways: A capital letter, such as A, B, or C, can denote a matrix. A lowercase letter enclosed in brackets, such as [a ij ] can denote a matrix. A general element in matrix A is denoted by a ij. This refers to the element in the ith row and jth column.

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Notations for Matrices (continued) A matrix of order has m rows and n columns. If m = n, a matrix has the same number of rows as columns and is called a square matrix.

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example: Matrix Notation Let a. What is the order of A? b. Identify a 12 c. Identify a 31

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Equality of Matrices Two matrices are equal if and only if they have the same order and corresponding elements are equal.

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Matrix Addition and Subtraction

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Example: Adding and Subtracting Matrices Perform the indicated matrix operations:

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Example: Adding and Subtracting Matrices Perform the indicated matrix operations:

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Properties of Matrix Addition

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Definition of Scalar Multiplication

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Scalar Multiplication If and find the following matrix: –6B

13. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Scalar Multiplication If and find 3A + 2B:

14. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Properties of Scalar Multiplication

15. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Example: Solving a Matrix Equation Solve for X in the matrix equation 3X + A = B, where and

16. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Matrix Multiplication

17. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Example: Multiplying Matrices Find AB, given and

18. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 Definition of Matrix Multiplication For the product of two matrices to be defined, the number of columns of the first matrix must equal the number of rows of the second matrix.

19. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 Example: Multiplying Matrices If possible, find the product:

20. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 Example: Multiplying Matrices If possible, find the product: The number of columns in the first matrix does not equal the number of rows in the second matrix. The product is undefined.

21. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21 Properties of Matrix Multiplication

22. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22 Example: Application Consider the triangle represented by the matrix Use matrix operations to move the triangle 3 units to the left and 1 unit down. coordinates of the vertices x-coordinates y-coordinates

23. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23 Example: Application (continued) We will subtract 3 from the x-coordinates (row 1) and subtract 1 from the y-coordinates (row 2).

24. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 24 Example: Application Consider the triangle represented by the matrix Use matrix operations to enlarge the triangle to twice its original perimeter. coordinates of the vertices x-coordinates y-coordinates

25. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 25 Example: Application (continued) We will multiply the original matrix by 2. This will enlarge the triangle to twice its original perimeter.

26. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 26 Example: Application Consider the triangle represented by the matrix Let Find BA. What effect does this have on the original triangle? coordinates of the vertices x-coordinates y-coordinates

27. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 27 Example: Application (continued) Multiplication by B reflects the original triangle over the x-axis.