1. Copyright © 2011 Pearson, Inc. 7.2 Matrix Algebra

2. Copyright © 2011 Pearson, Inc. Slide 7.2 - 2 What you’ll learn about Matrices Matrix Addition and Subtraction Matrix Multiplication Identity and Inverse Matrices Determinant of a Square Matrix Applications … and why Matrix algebra provides a powerful technique to manipulate large data sets and solve the related problems that are modeled by the matrices.

3. Copyright © 2011 Pearson, Inc. Slide 7.2 - 3 Matrix

4. Copyright © 2011 Pearson, Inc. Slide 7.2 - 4 Matrix Vocabulary Each element, or entry, a ij, of the matrix uses double subscript notation. The row subscript is the first subscript i, and the column subscript is j. The element a ij is the ith row and the jth column. In general, the order of an m  n matrix is m  n.

5. Copyright © 2011 Pearson, Inc. Slide 7.2 - 5 Example Determining the Order of a Matrix

6. Copyright © 2011 Pearson, Inc. Slide 7.2 - 6 Example Determining the Order of a Matrix

7. Copyright © 2011 Pearson, Inc. Slide 7.2 - 7 Matrix Addition and Matrix Subtraction

8. Copyright © 2011 Pearson, Inc. Slide 7.2 - 8 Example Matrix Addition

9. Copyright © 2011 Pearson, Inc. Slide 7.2 - 9 Example Matrix Addition

10. Copyright © 2011 Pearson, Inc. Slide 7.2 - 10 Example Using Scalar Multiplication

11. Copyright © 2011 Pearson, Inc. Slide 7.2 - 11 Example Using Scalar Multiplication

12. Copyright © 2011 Pearson, Inc. Slide 7.2 - 12 The Zero Matrix

13. Copyright © 2011 Pearson, Inc. Slide 7.2 - 13 Additive Inverse

14. Copyright © 2011 Pearson, Inc. Slide 7.2 - 14 Matrix Multiplication

15. Copyright © 2011 Pearson, Inc. Slide 7.2 - 15 Example Matrix Multiplication

16. Copyright © 2011 Pearson, Inc. Slide 7.2 - 16 Example Matrix Multiplication

17. Copyright © 2011 Pearson, Inc. Slide 7.2 - 17 Example Matrix Multiplication

18. Copyright © 2011 Pearson, Inc. Slide 7.2 - 18 Example Matrix Multiplication

19. Copyright © 2011 Pearson, Inc. Slide 7.2 - 19 Identity Matrix

20. Copyright © 2011 Pearson, Inc. Slide 7.2 - 20 Inverse of a Square Matrix

21. Copyright © 2011 Pearson, Inc. Slide 7.2 - 21 Inverse of a 2 × 2 Matrix

22. Copyright © 2011 Pearson, Inc. Slide 7.2 - 22 Determinant of a Square Matrix

23. Copyright © 2011 Pearson, Inc. Slide 7.2 - 23 Inverses of n  n Matrices An n  n matrix A has an inverse if and only if det A ≠ 0.

24. Copyright © 2011 Pearson, Inc. Slide 7.2 - 24 Example Finding Inverse Matrices

25. Copyright © 2011 Pearson, Inc. Slide 7.2 - 25 Example Finding Inverse Matrices

26. Copyright © 2011 Pearson, Inc. Slide 7.2 - 26 Example Finding Inverse Matrices

27. Copyright © 2011 Pearson, Inc. Slide 7.2 - 27 Properties of Matrices Let A, B, and C be matrices whose orders are such that the following sums, differences, and products are defined. 1. Community property Addition: A + B = B + A Multiplication: Does not hold in general 2. Associative property Addition: (A + B) + C = A + (B + C) Multiplication: (AB)C = A(BC) 3. Identity property Addition: A + 0 = A Multiplication: A·I n = I n ·A = A

28. Copyright © 2011 Pearson, Inc. Slide 7.2 - 28 Properties of Matrices Let A, B, and C be matrices whose orders are such that the following sums, differences, and products are defined. 4. Inverse property Addition: A + (-A) = 0 Multiplication: AA -1 = A -1 A = I n |A|≠0 5. Distributive property Multiplication over addition: A(B + C) = AB + AC (A + B)C = AC + BC Multiplication over subtraction: A(B – C) = AB – AC (A – B)C = AC – BC

29. Copyright © 2011 Pearson, Inc. Slide 7.2 - 29 Quick Review

30. Copyright © 2011 Pearson, Inc. Slide 7.2 - 30 Quick Review Solutions