1. Precalculus Lesson 7.2 Matrix Algebra

2. Quick Review

3. 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.

4. Matrix

5. Matrix Vocabulary Each element, or entry, aij, of the matrix uses double subscript notation. The row subscript is the first subscript i, and the column subscript is j. The element aij is the ith row and the jth column. In general, the orderof anm × n matrix is m×n.

6. Example Determining the Order of a Matrix

7. Matrix Addition and Matrix Subtraction

8. Example Matrix Addition

9. Example Using Scalar Multiplication

10. The Zero Matrix Example:

11. Additive Inverse

12. Example Using Additive Inverse

13. Matrix Multiplication

14. Example Matrix Multiplication

15. Identity Matrix

16. Inverse of a Square Matrix

17. Example Inverse of a Square Matrices Yes

18. Inverse of a 2 × 2 Matrix

19. Determinant of a Square Matrix Refer to text pg 583

20. Inverses of n× n Matrices An n × n matrix A has an inverse if and only if detA ≠ 0.

21. Example Finding Inverse Matrices

22. Properties of Matrices Let A, B, and C be matrices whose orders are such that the following sums, differences, and products are defined. 1. Commutative 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·In = In·A = A

23. 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-1A = In |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

24. Homework: Text pg588/589 Exercises #2, 4, 14, 20, 24, and 34