Chapter 6

1. Chapter 6 6.5-6.7 Matrices Please pick up a survey on the way in. You will be given time to complete it at the end of class.

2. 6.5 Matrices A matrix is a rectangular array of numbers called entries. Order (Size) of A: C1 C2 C3 R1 # Rows x # Columns R2 2 x 3 C1 C2 C3 3 x 3 R1 Square Matrix R2 Diagonal Entries R3 Remember the order of a matrix is always given Row x Column; This will be very important when we multiply matrices!

3. 6.5 Question #1 a) 2 x 3 and 3 x 3 b) 3 x 2 and 3 x 3 What type of matrix is C? What are the diagonal entries? What is the entry in row three column 2 of C?

4. 6.5 Augmented Matrices A matrix derived from a system of equations is called an augmented matrix. x y z 1 1 1 No y 1

5. 6.5 Example 1

6. 6.5 Example 2

7. 6.5 Example 2 If we divide the last row in part c by 16 we get: This is called a triangular matrix or the row-echelon form of a matrix.

8. 6.5 Example 2 We can easily solve the system from part c: x + 4y - z = -10 z = 1 -3y +10z = 7 x + 4(1) – 1 = -10 -3y +10(1) = 7 x + 4 – 1 = -10 -3y = -3 x + 3 = -10 y= 1 x = -13 (-7, 1, 1)

9. 6.5 Question # 2 Write the system of equations that corresponds to the given matrix: See A-4 in your text

10. 6.5 Row Operations

11. 6.5 Solving Systems w/ Matrices Get the matrix in triangular form. Back substitute.

12. 6.5 Example 3

13. 6.5 Example 3 -2 -2 -2 2 + 2 1 -2 -7 0 -1 -4 -5 1 4 5 0 2 8 10 + 0 -2 -1 -3_ 0 0 7 7

14. 6.5 Example 3 0 2 8 10 + 0 -2 -1 -3_ 0 0 7 7 x + y + z = -1 y + 4z = 5 x + 1 + 1 = -1 y + 4(1) = 5 y + 4 = 5 x + 2 = -1 x = -3 y = 1

15. 6.5 Example 4 -1 -1 5 -3 + 1 0 -2 1_ 0 -1 3 -2 -2 -2 10 -6 + 2 -1 -1 0_ 0 -3 9 -6

16. 6.5 Example 4 -2 -2 10 -6 + 2 -1 -1 0_ 0 -3 9 -6

17. 6.5 Example 4 y = 3z + 2 If z = -1, what solution do we have? (-1, -1, -1)

18. 6.5 Question # 3 Which of the following matrices represents an inconsistent system? 0 = 0 z = 0 0 = 1

19. 6.6 Equality of Matrices General m x n matrix For any two matrices A and B, A = B iff aij = bij for every entry in A and B.

20. 6.6 Example 5 FALSE FALSE CONDITIONAL

21. Scalar Multiplication means multiply every entry by a constant: 6.6 Addition, Subtraction & Scalar Multiplication Add corresponding entries. Subtract corresponding entries.

22. D 6.6 Example 6 Given: and find:

23. 6.6 Example 6 3 x 2 2 x 3

24. 6.6 Example 6 d. ½D

25. 6.6 Matrix Multiplication

26. 6.6 Example 7 2 x 2 2 x 2 Matches, we can do the multiplication Resulting matrix has order 2 x 2

27. 6.6 Example 7 2 x 2 3 x 2 Does not match

28. 6.6 Example 7 3 x 2 2 x 2 Matches, we can do the multiplication Resulting matrix has order 3 x 2

29. 6.6 Example 7 3 x 3 3 x 3 Resulting matrix has order 3 x 3 Matches, we can do the multiplication

30. 6.6 Example 7

31. 6.6 Properties of matrix Multiplication Note in general, matrix multiplication is not commutative; AB ≠ BA There are two exceptions…

32. 6.7 Identity Matrix The identity matrix is a square n x n matrix with 1’s along its diagonal and zeros everywhere else. If A is an n x n matrix then AIn = InA = A

33. 6.7 Example 8

34. 6.7 Inverse of a Matrix A must be square We’ll see later this is the determinant of A A matrix must be square to have an inverse, but not every square matrix has an inverse. A matrix that does not have an inverse is called a singular matrix.

35. 6.7 Finding A-1 for larger order matrices • By hand: • With a calculator: [A|I] [I|A-1] Row operations http://www.ncsu.edu/felder-public/kenny/papers/ti.html#MATRICES

36. 6.7 Solve Systems Using Inverses A is Coefficient Matrix X is variable matrix B is Constant Matrix A X B

37. 6.7 Question # 4 x = y = z =

38. 6.7 Question # 5 20 - 20

39. 6.7 Determinants det(A)

40. 6.7 Example 9

41. 6.7 Example 9

42. 6.7 Determinant of 3 x 3 Matrix We can use any row or any column to expand, but we must change our signs accordingly.

43. 6.7 Example 10 − + − + − −

44. Exit Survey Please complete the survey you were given on the way in and turn it in on your way out. Please put your survey in the correct crn pile.