MATRICES DEFINITION A rectangular array of numeric or

MATRICES

DEFINITION • A rectangular array of numeric or algebraic quantities subject to mathematical operations. • Something resembling such an array, as in the regular formation of elements into columns and rows.

DEFINITION •

EQUAL MATRICES •

YOU TRY… •

ADDING AND SUBTRACTING MATRICES • In order to add or subtract matrices, the must have the same order.

ADDING MATRICES •

SUBTRACTING MATRICES •

SOLVING MATRIX EQUATIONS FOR X •

INVERSES

IDENTITY ELEMENT •

DETERMINANT •

MULTIPLICATION

CHECK FIRST • Use the order of the matrix to check whether or not we can multiply them: • The number columns of the first matrix must match the number of rows in the second matrix. • The number of rows in the first matrix paired with the number of columns in the second matrix gives us the dimensions of the product matrix.

They must match. Dimensions: 2 x 3 The dimensions of your answer. 3 x 2

EXAMPLES •

HOW DO WE DO IT? •

WHAT’S THE PATTERN? •

EXAMPLES

Examples: 2(3) + -1(5) 2(-9) + -1(7) 2(2) + -1(-6) 3(3) + 4(5) 3(-9) + 4(7) 3(2) + 4(-6)

Dimensions: 2 x 3 2 x 2 *They don’t match so can’t be multiplied together. *

*Answer should be a 2 x 2 0(4) + (-1)(-2) 0(-3) + (-1)(5) 1(4) + 0(-2) 1(-3) +0(5)

SYSTEMS OF EQUATIONS APPLICATIONS

STEPS TO CREATING AN EQUATION FROM CONTEXT: 1. Read the problem statement 2. Identify the known quantities 3. Identify the unknown variables 4. Create an equation using the known quantities and the variables you found. Keep in mind… q In systems of equations, you will have 2 or 3 equations (hence the “system” part). Therefore, you will have to decide which quantities and variables belong together.

EXAMPLE • The admission fee at a small fair is $1. 50 for children and $4. 00 for adults. On a certain day, 2200 people enter the fair and $5, 050 is collected. How many children and how many adults attended?

EXAMPLE 2 • Two small pitchers and one large pitcher can hold 8 cups of water. One large pitcher minus one small pitcher constitutes 2 cups of water. How many cups of water can each pitcher hold?

EXAMPLE 3 • A test has twenty questions worth 100 points. The test consists of True/False questions worth 3 points each and multiple choice questions worth 11 points each. How many multiple choice questions are on the test?

SOLVING SYSTEMS USING INVERSE MATRICE S

Solve the system using inverse matrices 3 x + 2 y = 7 4 x - 5 y = 11 You can use the inverse of the coefficient matrix to find the solution. Set up a matrix equation to find the solution. The formula to find the solution is:

Solve the system using inverse matrices 2 x - 4 y = 9 3 x - 2 y = 1 Set up a matrix equation to find the solution. The formula to find the solution is:

Solve the system using inverse matrices x + 4 y = 8 2 x - 2 y = -6 Set up a matrix equation to find the solution. The formula to find the solution is:

Solve the system using inverse matrices The formula to find the solution is: Set up a matrix equation to find the solution.

VERTEX-EDGE GRAPHS

WHAT IS A VERTEX-EDGE GRAPH? • A collection of points, some of which are joined by line segments or curves • Examples:

VERTEX • A point on the graph.

EDGE • A line segment or curve connecting the vertices of a graph.

IN THE REAL-WORLD • Vertices may represent things such as people or places. • Edges may represent connections such as roads or relationships. What are the edges, and what do they represent? What about the vertices?

COMPLETE GRAPH • A graph in which every vertex is adjacent to every other vertex. • Which of these is complete? A B

DIGRAPH • A directed vertex edge graph

DEGREE OF THE VERTEX • The number of edges that enter a vertex. What is the degree of vertex A? 2 What is the degree of vertex C? 4

REAL WORLD EXAMPLE • The vertex edge graph below represents five people: Bob (B), Dustin (D), Mike (M), Sue (S) and Tammy (T). • An edge connecting two vertices indicates that those two people have a class together.

WHO HAS A CLASS WITH MIKE? Tammy & Sue

WHO DOES NOT HAVE A CLASS WITH BOB? Tammy & Mike

USING MATRICES TO REPRESENT A VERTEX-EDGE GRAPH • We can use an adjacency matrix to represent the vertex-edge graph. • Step 1: Create a matrix listing all vertices in the row and column. • Step 2: Fill in the matrix listing the number of relationships between the two points. • If they share an edge, there will be a “ 1” • If there is no relationship, there will be a “ 0”

CREATE A MATRIX USING THE FOLLOWING VERTEX-EDGE GRAPH:

CREATE A MATRIX USING THE FOLLOWING VERTEX-EDGE GRAPH: R S V T U

DRAWING A VERTEX-EDGE GRAPH • Use the following matrix to create the vertexedge graph that corresponds.

DRAWING A VERTEX-EDGE GRAPH • Use the following matrix to create the vertexedge graph that corresponds.

EXTENDED RELATIONSHIPS • A railway serves four cities: Harrisburg, Baltimore, Philadelphia and Atlantic City. Trains travel between Harrisburg and Baltimore, Harrisburg and Philadelphia, and Philadelphia and Atlantic City. • Draw a vertex edge graph and it’s adjacency matrix to represent this situation.

RAILWAY CONTINUED… •

SUMMARY 3 -2 -1 • On a separate sheet of paper to turn in, list: • 3 vocabulary words you have learned and their relationship to the vertex-edge graph • 2 reasons to use a vertex-edge graph • 1 real-world example of a vertex-edge graph