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Matrices. Chapter 6. Warmup – Solve Trig Equations. Unit 5.3 Page 327 Solve 4sinx = 2sinx + √2 2sinx = √2 Subtract 2sinx from both sides Sinx = √2/2 Divide 2 from both sides Work on the following problems Unit 5.3 Page 331 Problems 1, 3, 4, 9, 10. Chapter 6: Matrices. - PowerPoint PPT Presentation
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Matrices
Chapter 6
Warmup – Solve Trig Equations
• Unit 5.3 Page 327
Solve 4sinx = 2sinx + √2
1. 2sinx = √2 Subtract 2sinx from both sides
2. Sinx = √2/2 Divide 2 from both sides
A. Work on the following problems
Unit 5.3 Page 331 Problems 1, 3, 4, 9, 10
Chapter 6: Matrices
What are matrices? Rectangular array of mn real or complex numbers arranged in m horizontal rows and n vertical columns
rows 2 3 4
7 1 5
6 1 0
columns
Matrices: Why should I care?
1. Matrices are used everyday when we use a search engine such as Google:
Example: Airline distances between citiesLondon Madrid NY Tokyo
London 0 785 3469 5959Madrid785 0 3593 6706NY 3469 3593 0 6757Tokyo 5959 6706 6757 0
Quick Review
Unit 6.1 Page 372
Problems 1 - 4
Write an Augmented Matrix
Unit 6.1 Page 366 Guided Practice 2a
w + 4x + 0y + z = 2 1 4 0 1 2
0w + x + 2y – 3z = 0 0 1 2 -3 0
w + 0x – 3y – 8z = -1 1 0 -3 -8 -1
3w +2x + 3y + 0z = 9 3 2 3 0 9
Page 372 Problems 9 - 14
Row Echelon Form
Objective: Solve for several variables1 a b c0 1 d e0 0 1 fThe first entry in a row with nonzero entries
is 1, or leading 1For the next successive row, the leading 1 in
the higher row is farther to the left than the leading 1 in the lower row
Row Echelon form
Unit 6.1 Page 372 Problems 16 - 21
Gauss-Jordan Elimination
• How do I solve for each variable? x + 2y – 3z = 7 -3x - 7y + 9z = -12 2x + y – 5z = 8Augmented1 2 -3 7-3 -7 9 -122 1 -5 8
Gauss-Jordan Elimination
1. Use the following steps on your graphing calculator2nd →matrixEditSelect AChoose Dimensions (row x column)Enter numbers2nd → quit2nd →matrixMathRref (reduced reduction echelon form)2nd Matrix → select
Gauss-Jordan Elimination
Unit 6.1 Page 372
Problems 24 - 28
Multiplying with matrices
• 3 Types
a. Matrix addition (warm-up)
b. Scalar multiplication
c. Matrix multiplication
Adding matrices
1. Only one rule, both rows and columns must be equal
1. If one matrix is a 3 x 4, then the other matrix must also by 3 x 4
Which of the following matrix cannot be added?
A B C D2 x 4 7 x 8 10 x 11 14 x 123 x 4 7 x 8 10 x 11 14 x 12
Scalar Multiplication
• {-2 1 3} 4
• -6 = { (-2)4 + 1(-6) + 3(5) } = {1}
• 5
Multiplying Matrices
• In order for matrices to be multiplied, the number of columns in matrix A, must equal the number of rows in matrix B.
• Matrix A Matrix B
• 3 x 2 2 x 4• equal
• New proportions
Multiplying Matrices
Procedures – row times column A B 3 -1 -2 0 64 0 3 5 1
3 (-2) + (-1)3 3(0) + -1(5) (3)(6) + (-1)14(-2) +(0)3 4(0) + 0(5) 4(6) + (0)1
Answer -9 -5 17-8 0 24
Unit 6.2
• Page 383 Problems 1 – 8
• 1. Determine if the matrices can be multiplied, then computer A x B
Unit 6.2
• Problems 1 – 8, 19 - 26