Review Solve the system of equations. 1 2, -1, 1

Matrices 2

Definition Some Words: One: Matrix More than one: Matrices Definition: In Mathematics, matrices are used to store information. This information is written in a rectangular arrangement of rows and columns. Each entry, or element, of a matrix has a precise position and meaning. 3

Example Food shopping online: people go online to shop for items and have them delivered to their homes. Cartons of eggs, bread, packets of vegetables, bags of rice, packets of fish were ordered online and the people left their address for delivery. A selection of orders may look like this: 4

Example Order Address Carton of eggs bread vegetablesricefish 10 Kros Road Usmar St High St Ofar Rd

Example The dispatch people will be interested in the numbers: This is a 4 by 5 matrix 4 rows 5 columns 6

Definition A matrix is defined by its order which is always number of rows by number of columns 7 RXC 2 rows 3 columns 2 X 3 matrix

Exercise Consider the network below showing the roads connecting four towns and the distances, in km, along each road. 8 A 14 C D B (i) Write down the information in matrix form. (ii) What is the order of the matrix?

Solution (i) This information could be put into a table: 9 kmABCD A B C D to from

Solution and then into a matrix: 10 (ii) order: R X C= 4 X 4 matrix. This is called a square matrix.

Definition A square matrix has the same number of rows as columns. Its order is of the form M x M. Examples: 11 2 X 2 square matrix 3 X 3 square matrix

Definition The transpose of a matrix M, called M T, is found by interchanging the rows and columns. Example: M = row column

Definition Equal Matrices: Two matrices are equal if their corresponding entries (elements) are equal. Example: If 13 a = 10 c = 4d = 8 b = -2 =

Definition Entries, or elements, of a matrix are named according to their position in the matrix. The row is named first and the column second. Example: entry a 23 is the element on row 2, column 3. Example: here are the entries for a 2 x 2 matrix. 14

Example In the following matrix, name the position of the colored entry. (i) Remember: row first a2a2 Column second row 2 column 1 The entry is a 21

Example In the following matrix, name the position of the colored entry. (ii) 16 c d e f o p q r row 1, column 3 The entry is a 13

Example In the following matrices, identify the value of the entry for the given position. 17 a 32 a 24 row 3, column 2 = 5 row 2, column 4 = 2

Example 1-1a College Kaitlin wants to attend one of three Iowa universities next year. She has gathered information about tuition (T), room and board (R/B), and enrollment (E) for the universities. Use a matrix to organize the information. Which universitys total cost is lowest? Iowa State University: T - $3132 R/B - $4432 E - 26,845 University of Iowa: T - $3204 R/B - $4597 E - 28,311 University of Northern Iowa: T - $3130 R/B - $4149 E - 14,106

Example 1-1b Organize the data into labeled columns and rows. ISU UI UNI TR/BE Answer:The University of Northern Iowa has the lowest total cost.

Example 1-1c Dining Out Justin is going out for lunch. The information he has gathered from the two fast-food restaurants is listed below. Use a matrix to organize the information. When is each restaurants total cost less expensive? Burger ComplexLunch Express Hamburger Meal $3.39 Hamburger Meal $3.49 Cheeseburger Meal $3.59 Cheeseburger Meal $3.79 Chicken Sandwich Meal $4.99 Chicken Sandwich Meal $4.89

Example 1-1d The Burger Complex has the best price for hamburgers and cheeseburgers. Lunch Express has the best price for chicken sandwiches. Answer: Burger Complex Lunch Express Hamburger Meal Cheese- burger Meal Chicken Sandwich Meal

Example 1-2a State the dimensions of matrix G if 4 columns 2 rows Answer:Since matrix G has 2 rows and 4 columns, the dimensions of matrix G are 2 4.

Example 1-2b State the dimensions of matrix G if Answer:3 2

Example 1-3a Solvefor x and y. Since the matrices are equal, the corresponding elements are equal. When you write the sentences to solve this equation, two linear equations are formed.

Example 1-3b Second equation This system can be solved using substitution. Substitute 3x – 2 for y. Distributive Property Add 4 to each side. Divide each side by 7.

Example 1-3c To find the value for y, substitute 1 for x in either equation. Substitute 1 for x. First equation Simplify. Answer:The solution is (1, 1).

Example 1-3d Solvefor x and y. Answer: (2, 5)

Review State the dimensions of each matrix. Then identify the position of the circled element in each matrix. 28

Review Solve the equation. 29