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 𝑎 11 𝑎 12 𝑎 13 𝑎 21 𝑎 22 𝑎 23 𝑎 31 𝑎 32 𝑎 33 If you have 𝑎 𝑛𝑚 : n is the row number m is the column number.

Equal Matrices 𝑎 𝑏−2 2𝑐 𝑑+4 = 5 12 22 8 𝑎=5 𝑏−2=12 2𝑐=22 𝑑+4=8

You Try… 3𝑎 𝑏−9 𝑐 2 𝑑 0 ℎ+6 2𝑓+1 ℎ 2 2 = 12 −3 16 2 0 29 19 25 2

Adding and Subtracting Matrices In order to add or subtract matrices, the must have the same order.

Adding Matrices 2 3 4 5 + 3 7 6 9 You Try… −2 3 4 −7 + −1 7 4 6

Subtracting Matrices 2 3 4 5 − 3 7 6 9 You Try… −2 3 4 −7 − −1 7 4 6

Solving Matrix Equations for X 𝑋− 8 15 7 5 = 9 −9 6 0 You Try… 𝑋+ 4 7 8 11 = −3 0 9 4

Inverses

Identity Element In the identity matrix, there is a series of ones along the diagonal with zeros in all the other places. 2𝑥2 Matrix 1 0 0 1 3𝑥3 Matrix 1 0 0 0 1 0 0 0 1

The notation for determinant is 𝐴 If the determinant is 0, the matrix has no inverse! 2𝑥2 Matrix 𝐴 =𝑎𝑑−𝑏𝑐

Notation: 𝐴 −1 = 1 𝐴 𝑑 −𝑏 −𝑐 𝑎 = 1 𝑎𝑑−𝑏𝑐 𝑑 −𝑏 −𝑐 𝑎 Inverse of a 2𝑥2 Matrix Notation: The inverse of matrix 𝐴 = 𝐴 −1 𝐴 −1 = 1 𝐴 𝑑 −𝑏 −𝑐 𝑎 = 1 𝑎𝑑−𝑏𝑐 𝑑 −𝑏 −𝑐 𝑎

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: 3 x 2 2 x 3 The dimensions of your answer.

Examples 2𝑋3 ∗3𝑋2 4𝑋1 ∗4𝑋1 2𝑋6 ∗6𝑋4 1 2 3 4 5 6 7 8 9 1 2 3

How do we do it? 𝐴= 1 2 3 4 ; 𝐵= 5 6 7 8 To multiply matrices, we multiply the ROWS of the first matrix by the COLUMNS of the second. 𝐴𝐵= 1 2 3 4 5 6 7 8

𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ = 𝑎𝑒+𝑏𝑔 𝑎𝑓+𝑏ℎ 𝑐𝑒+𝑑𝑔 𝑐𝑓+𝑑ℎ What’s the Pattern? 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ = 𝑎𝑒+𝑏𝑔 𝑎𝑓+𝑏ℎ 𝑐𝑒+𝑑𝑔 𝑐𝑓+𝑑ℎ

examples Do problems from classwork: 3, 4, 12, 19, 26

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)

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

2 x 2 2 x 2 *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: Read the problem statement Identify the known quantities Identify the unknown variables Create an equation using the known quantities and the variables you found. Keep in mind… 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?

Using Inverse Matrices Solving Systems Using Inverse Matrices

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

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

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

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? B A

Digraph A directed vertex edge graph

2 4 Degree of the Vertex The number of edges that enter a vertex. What is the degree of vertex A? What is the degree of vertex C? 2 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:

Drawing a Vertex-Edge Graph Use the following matrix to create the vertex-edge graph that corresponds. 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 1

Drawing a Vertex-Edge Graph Use the following matrix to create the vertex-edge 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… If we take 𝐴 2 , we can see the relationship that is formed by using 2 trains. What relationship would 𝐴 3 give?

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