Grade 7/8 Math Circles November 1 & 2, The Matrix

Transcription

1 Faculty of Mathematics Waterloo, Ontario N2L 3G Centre for Education in Mathematics and Computing Grade 7/8 Math Circles November & 2, 26 The Matrix Introduction to Matrices A matrix is a rectangular arrangement of numbers. We can talk about one matrix or several matrices. Matrices are often used to represent information without labels. For example, suppose you and a friend each have a box of Smarties and you each want to count all your red and blue Smarties. This information is represented in the table below. Red Blue You 5 4 Friend

2 Example Suppose Farm A has 8 pigs and 4 cows and Farm B has 3 pigs and 9 cows. Represent this information in a matrix. Pigs Cows Farm A 8 4 Farm B Now we know how to translate information from a table to a matrix. Let s talk about the dimensions of a matrix. So far, we have only seen 2 2 matrices. In general, an m n matrix has m rows and n columns. Matrix A, in the example below, is a 2 3 matrix since there are two rows and three columns. } 4 6 A 2 rows }{{} 3 columns If a matrix has m rows and m columns, it is an m m matrix or a square matrix. The matrix below is an example of a 2 2 matrix which is a square matrix Example What are the dimensions of the following matrices? (a) 2 3 matrix (b) matrix 7 2 The numbers inside a matrix are called elements. To generalize, we can label each element using variables in our matrices. For example, a b a b c a a b c d c d a b c d e f d e f b e f g h i c 2

3 Life With Matrices How exactly are matrices used in the world? Matrices have many real-life applications. Here are a few examples: Cryptography - Matrices can be used to encrypt and decrypt secret messages. The sender encrypts the message using an encoding matrix and the receiver decrypts it using a decoding matrix. encoding matrix message encrypted message decoding matrix encrypted message message Economics - Businesses use matrices to study stock market trends to earn a larger profit and to lower their losses Revenue Costs P rof it November Google - Ever wonder how Google always knows what you search for? Or wonder how they determine what sites to direct you to? Google uses a page ranking method that uses matrices! 3

4 Matrix Operations Matrix operations are quite interesting. They do not follow the same rules as numbers do and have their own ways of performing our usual operations. Matrix Addition Matrix addition is fairly straightforward. To perform matrix addition, we add together elements that are in the same position in each matrix. Suppose we want to add the two 2 2 matrices: a b e f + c d g h a + e b + f c + g d + h Note that when adding matrices together, all matrices must have the same dimensions! We were able to add the matrices above together because both matrices are 2 2. Example Add the following matrices: Example 2 Go Raptors! You re watching a Toronto Raptors basketball game. In the first quarter, Kyle Lowry scored points with block and 2 assists. DeMar DeRozan scored 3 points with blocks and assist. Jonas Valanciunas scored 9 points with 3 blocks and assists In the second quarter, Lowry scored 5 points with blocks and 3 assists. DeRozan scored points with blocks and 2 assists. JV scored 4 points with 2 blocks and assist. How many points, blocks, and assists does each player have by halftime? By halftime, Lowry scored 26 points with block and 5 assists, DeRozan scored 24 points with blocks and 3 assists, and JV scored 23 points with 5 blocks and assist. 4

5 Matrix Subtraction Matrix subtraction works the same way as addition! To perform matrix subtraction, we subtract elements that are in the same position in each matrix. a b e f c d g h a e b f c g d h Similar to matrix addition, matrices need to have the same dimensions in order to perform matrix subtraction. Example Subtract the following matrices: Example 2 Bart and Lisa get a weekly allowance of $5 and they are competing to see who can save the most money until Wednesday. On Monday, Bart and Lisa do not spend anything On Tuesday, Bart buys a snack at the vending machine for $.25 On Wednesday, Bart loses $.25 in a bet with his friends and Lisa spend $.7 on bubble gum On Wednesday, who saved the most amount of money? Bart or Lisa? By Wednesday, Bart has $3.5 and Lisa has $3.3, so Bart saved the most money. 5

6 Scalar Multiplication A scalar is a any number outside of the matrix. To perform scalar multiplication, we multiply each element by the same number. Suppose we want to calculate the following: a b k c d ka kb kc kd Example Calculate the following: Example 2 Yesterday, there are were 45 kids, 72 teenagers, and 3 adults who played Laser Tag and 22 kids, 55 teenagers, and 53 adults who played the Escape Room. Today, there were twice as many people because of a coupon found in the flyers. How many kids, teenagers, and adults played Laser Tag? How many played the Escape Room? Today, 9 kids, 44 teenagers, and 62 adults played Laser Tag and 44 kids, teenagers, and 6 adults played the Escape Room. 6

7 Matrix Multiplication Here, we are multiplying a matrix by another matrix. To perform matrix multiplication, we multiply each element of a row from one matrix by the corresponding element of the column from the other matrix and add the products together. Below, see what happens when we multiply two 2 2 matrices together. a b e f c d g h ae + bg af + bh ce + dg cf + dh Example Calculate the following: (a) (b) (c) (4) + 5(2) (3) + 5() 2(4) + 7(2) 2(3) + 7() (7) + 2(2) + () 3() + 2(5) + (4) (7) + 3(2) + 4() 5() + 3(5) + 4(4) (2) + 4(4) + 2() (2) + 5(4) + 6() (2) + 9(4) + 3() There are a few things we need to take note of: 53 Suppose A is an m n matrix and B is a p q matrix.. Matrices A and B can be multiplied together only if n p (i.e. # of columns in A # of rows in B) 2. The product of the two matrices, A and B, will be an m q matrix (i.e. AB will be an m q matrix.) 7

8 Problem Set Solutions. Construct a matrix for the following scenarios: Alternate solution: Switch the labels for the rows and columns. (a) We are doing a class vote on which one is better: books or movies. For books, there were votes from the Grades 7s and 6 votes from the Grade 8s. For movies, there were 8 votes from the Grade 7s and votes from the Grade 8s. Books Movies Grade 7 8 Grade (b) It s the season for apple ciders and hot chocolate drinks! On Friday, Tim Horton s sells 25 apple ciders, 32 hot chocolates, and 28 white hot chocolates. On Saturday, they sell 53 apple ciders, 55 hot chocolates, and 46 white hot chocolates. Sunday, they sell 45 apple ciders, 45, hot chocolates, and 47 white hot chocolates. On Apple Cider Hot Choco White Hot Choco Friday Saturday Sunday Add or subtract the following matrices: A B 5 6 C (a) A + B (b) B C (c) A + B C (d) B + C A

9 3. Multiply the following matrices: 2 4 D (a) DE (b) EF E 9 F (c) F D (d) F DE Calculate the following given matrices G and H: 3 G (a) 3G + H H (b) 5G 4H When multiplying numbers, we can say In general, a b b a where a and b is any number. This is called the commutative property of numbers. Is this property true for matrices? Answer the following questions to find out. Suppose we have the following matrices: (a) Calculate the following: A B 8 5 i. AB ii. BA 6 47 AB BA (b) Does AB BA? Is the commutative property true for matrices? No. AB BA because matrix AB has different dimensions than matrix BA. Thus, the commutative property is not true for matrices. 9

10 6. Suppose CD E. What are the missing elements in matrix C? C D E Suppose D 2E F. What are the missing elements in matrices D, E and F? D E F Suppose 4M + NP Q. What is the matrix M? N P 8 3 Q M If you multiply a 3 6 matrix by a 6 2 matrix, what are the dimensions of the product? 3 2. What is another application of matrices in real life? (Do some research!) Answers may vary. Matrices are used in a number of different things. For example,... Physics: Matrices are used in the study of electrical circuits, quantum mechanics, and optics Robotics: The base elements of robot movements are matrices

11 . An electronics company called Alpha wants to determine which of their phone products made the most profit (or made the most money). In the last month, Phone A earned $5, and Phone B earned $2, in total Phone A cost $, and Phone B cost $75, to manufacture (Hint: The equation to calculate profit is Revenue Cost P rof it where revenue is the amount of money earned, cost is the amount of money spent, and profit is the amount of money leftover.) (a) How much profit did Alpha make in the last month total? 5,, Revenue Cost 2, 75, 5, Revenue Cost 25, Alpha made a total profit of $5, + $25, $75, in the last month from Phone A and Phone B combined. (b) Which phone product made the most profit in the last month? Phone A 2. During recess, T.J., Vince, and Mikey decide to combine their piggy bank money to buy $. worth of bubble gum after school at the convenience store. T.J. has 6 dimes, quarters, and 2 loonies, Vince has 2 dimes, and 7 quarters, Mikey has 8 dimes, 6 quarters, and loonie. Mikey, the kind soul that he is, buys a juice box for a thirsty first grader. It costs him $.25. Do they have enough money to buy $. worth of bubble gum? Dimes Quarters Loonies T.J. 6 2 Vince 2 7 Mikey Altogether, T.J., Vince, and Mikey have $5. + $ $3.3 $.45 but Mikey bought a juice box for $.25 so, in total they have $.45 - $.25 $.2. Yes, they have just enough money to buy all their bubble gum.

12 *3. Jean, Logan, and Scott all go to Professor X s School for Mutants. It s the end of the year and they all want to calculate their final grades for their Mutant History class. Their grade is made up of three tests: Test A, Test B, and Test C. Jean received an 85% on Test A, 8% on Test B, and 86% on Test C Logan received a 74% on Test A, 64% on Test B, and 88% on Test C Scott received an 77% on Test A, 92% on Test B, and an 74% on Test C Suppose Test A is worth 2% of their final grade, Test B is worth 3%, and Test C is worth 5%. What are the final grades in class for Jean, Logan, and Scott? Start by creating two matrices: one for the test grades of Jean, Logan, and Scott, and another matrix containing the final grade values of each test. Test A Test B Test C Jean Logan Scott Change the test grades into decimals since you want to calculate how many marks of their final grade a student receives given their test grade. (ex. Jean s Test A grade is worth marks of her final grade). Grade Value Test A 2 Test B 3 Test C (2) +.8(3) +.86(5) (2) +.64(3) +.88(5) (2) +.92(3) +.74(5) Jean 84 Logan 78 Scott 8 Thus, Jean s, Logan s, and Scott s final grades are 84%, 78%, and 8% respectively. 2

13 **4. Hill Cipher (Cryptography) Suppose we number the alphabet from to 25. (i.e. A, B, C 2,...). As an example, consider the message FRIEND as a 3 2 matrix and the 3 3 key matrix (or encoding matrix) below: F FRIEND R I E 5 4 N 7 3 D Key matrix To encrypt the message, multiply the key matrix by the message. For the resulting matrix, subtract 26 from each element repeatedly until the element is less than I C Z B B ICZBBC The encrypted message is ICZBBC. To decrpyt a message, you need another key matrix. Multiply this key matrix by the encrypted message matrix to determine the actual message. (Note that all elements in every matrix must be a number from to 25.) (a) Encrypt the following message given the key matrix: HAPPINESS 2 5 Key matrix HAPPINESS 8 8 C LDSTJISEI HAPPINESS LDSTJISEI (b) Decrypt the following encrypted message given the key matrix: YKVJ Key matrix YKVJ MATH YKVJ MATH 3

14 ***5. Google Page Rank Method Google determines which webpages to display first based on how often a webpage is visited using a combination of different mathematical concepts including matrices. At this point, the page rank method is quite complex to explain but if you are up for a serious challenge, research Google Page Rank. For now, to simplify, the following equation is used to rank webpages: G d L + ( d) N e L is the matrix that tell us whether there is a link between webpages (i.e. If there is a link between A and B, the matrix element is. Otherwise, it is.) d is a number between and, called the damping factor, which ensures that we will not be calculating the page rank infinitely (i.e. At some point, you stop clicking other webpages) N is the number of webpages e is a ones matrix with the same dimensions as G and L (i.e. All matrix elements are ) G is the updated matrix L or our rank matrix p is the probability matrix that a page will be clicked After calculating G, R G p R is our final matrix with the final ranking of all the webpages. The page with the highest ranking is displayed first on your Google search. 4

15 (a) For webpages A, B, and C, determine rank matrix G. d.85 L A B C A B N 3 e C G G (b) Given the probabiliy matrix p, what are the page rankings of webpages A, B, and C? (i.e. Determine matrix R.) p Note that in probability matrix p, webpage A has a 5% probability of being clicked, webpage B has a 5% probability, and webpage C has a 35% probability R Webpage A is ranked st and webpages B and C are both ranked 2 nd. 5