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