Warm-up 1.2

1. Warm-up 1.2 • Find matrix G if G = [0 2] [0 -3] [-1 0] [1 3] + [-1 4] - [2 -2] [2 1] [2 2] [-3 1]

2. Reminders:Sorry! I’ll be absent tomorrow for a training I have to go to. I will leave specific sub plans. All work must be turned in (I will come by the school and pick up work to grade it). If you have any questions, comment on the website or leave a note on your paper *What level are you?

10. §1.2: Organizing Data & Matrices LEQ: How can statistical data be organized into matrices? Uses for matrices…

11. Topics • Organizing Statistical Data • What is the difference between a 2x3 matrix and a 3x2 matrix? • Presenting Data from a table in a matrix • Elements, rows, columns (row, column) • Equal matrices = iff same dimensions and corresponding elements are equal • Solving matrices with algebra involved

13. Adding and Subtracting Matrices LEQ: How do you add, subtract, and multiply matrices? • Matrix addition/subtraction – add/subtract corresponding elements • Explain why you add matrices only if they have the same dimensions. • Is matrix subtraction commutative? Why? • Matrix equation – addition/subtraction properties of equality

14. 10.3: Matrix Multiplication How are the dimensions of a matrix related to the ability to multiply matrices? Scalar Multiplication (Scalar) You can also multiply two matrices. Method: • Multiply the elements of each row of the first matrix by the elements of the first column of the second matrix. • Add the products.

15. Use scalar Multiplication

34. Assignment Practice 3.1 – 3.2 Mixed Exercises Selected Problems?

35. Warm-up 1/11/08 Find the product, if possible. If not, possible, write produce undefined. • [2] [3 1 -2] [1] • [1] [1 0] [2] [0 1] • [-1 0 -2] [1] [2 3 1] [3] [2]

36. Identity and Inverse Matrices LEQ: How would you describe the identity matrix and its uses? A square matrix is a matrix with the same number of columns as it has rows. The Identity matrix is a square matrix with 1’s along the diagonal and 0’s everywhere else.

37. Inverse Matrix • IF X is the inverse matrix of A, then AX = I (A times its inverse = the identity) Not all matrices have inverses. If detA = 0, then A does not have an inverse. If detA ≠ 0, then A does have an inverse.

38. If A = [ a b] [ c d] Then, A-1 = *swap a & c, make b,d neg. 1 [c -b] (ad – bc) [a -d] You can also use inverse matrices to solve matrix equations. [0 -4] X = [0] [0 -1] [4] For this problem A-1 does not exist, so you cannot solve the problem.

39. Examples • 3-6 Worksheet

40. Solving Systems of LinearEquations in Three VariablesUsing the Elimination Method • Note that there is more than one way that you can solve this type of system.  Elimination (or addition) method is one of the more common ways of solving the problem by algebra, so I choose to show it this way.

41. Step 1: Simplify and put all three equations in the form Ax + By  + Cz = D if needed . • This would involve things like removing ( ) and  removing fractions. • To remove ( ): just use the distributive property.  • To remove fractions: since fractions are another way to write division, and the inverse of divide is to multiply, you remove fractions by multiplying both sides by the LCD of all of  your  fractions.

42. Step 2: Choose to eliminate any one of the variables from any pair of equations. • This works in the same manner as eliminating a variable with two linear equations and two variables • At this point, you are only working with two of your equations.  In the next step you will incorporate the third equation into the mix. 

43. Looking ahead, you will be adding these two equations together.  • Make sure one of the variables cancels out. • It doesn't matter which variable you choose to drop out. • For example, if you had a 2x in one equation and a 3x in another equation, you could multiply the first equation by 3 and get 6x and the second equation by -2 to get a -6x.  So when you go to add these two together they will cancel out.

44. Step 3:  Eliminate the SAME variable chosen in step 2 from any other pair of equations, creating a system of two equations and 2 unknowns. • Basically, you are going to do another elimination step, eliminating the same variable we did in step 2, just with a different pair of equations.  • Follow the same basic logic as shown in step 2

45. Step 4:  Solve the remaining system found in step 2 and 3, just as if it is a system of 2 equations in 2 variables • After steps 2 and 3, there will be two equations and two unknowns which is just a system of 2 equations and 2 variables. • You can use any method you want to solve it. • When you solve this system that has two equations and two variables, you will have the values for two of your variables.

46. Remember that if both variables drop out and you have a FALSE statement, that means your answer is no solution.  • If both variables drop out and you have a TRUE statement, that means your answer is infinite solutions, which would be the equation of the line.

47. Step 5:  Solve for the third variable. • If you come up with a value for the two variables in step 4, that means the three equations have one solution.  Plug the values found in step 4 into any of the equations in the problem that have the missing variable in it and solve for the third variable.

48. Step 6:  Check. • You can plug in the proposed solution into ALL THREE equations.  If it makes ALL THREE equations true then you have your solution to the system.  • If it makes at least one of them false, you need to go back and redo the problem.

49. There are three possible outcomes that you may encounter when working with these systems: • one solution • no solution • infinite solutions

50. Solve for x, y, and z -3x + 8y + 1z = -18 -2x – 6y – 9z = -15 -6x + 5y + 3z = 13 (-3,-4,5)