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Hwa Chong Institution Unit 5.1 Matrices - Matrix Representation Name: ________________________ ( ) Class: _________ Date: ________ Matrix Multiplication and Word Problems Objectives: Worksheet 3 1. Solve word problems using matrices 2. Perform matrix multiplication on small order matrices I Matrix Multiplication Visit the following website http://www.math.ncsu.edu/ma107/114/_c1d2a.mov to view the video on matrix multiplication. In summary, 1. Two matrices, A and B can be multiplied together iff the number of __________ in A is equal to the number of __________ in B. 2. The multiplication of two matrices, A and B, will give rise to a matrix AB with the number of __________ of A and the number of __________of B, i.e. Amn Bn p ABm p . 3. The product of two matrices, A and B, will give rise to a matrix AB, whose element in the ith row and jth column is the sum of the products formed by multiplying each element in the ith row of A by the corresponding element in the jth column of B. Practice 1. Compute the products AB and BA for the following, if possible: (i) 1 2 1 4 A , B 3 1 2 0.5 2 3 2 0.5 1 , B 1 2 1 3 4 (ii) A 1 1 1 2 2 2 (iii) A , B 2 0 0 4 0 3 3 4 II Laws of Matrix Multiplication 1. Given that A 3 1 2 5 1 0 , B and C , find (i) A( BC ) , 4 0 1 3 2 3 Note: 2. (ii) ( AB)C . A( BC ) ( AB)C , this question illustrates that matrix multiplication is associative. 2 1 2 4 1 0 , B and C , find 1 0 2 3 0 3 Given that A (i) Note: A( B C ) (ii) AB AC (iii) ( B C ) A (iv) BA CA A( B C ) AB AC and ( B C ) A BA CA , this illustrates that matrix multiplication is distributive over addition. SBGE/MA/SEC1/BEN/MAT/WS3 1 3. 2 1 4 1 , find and C 1 3 0 0 Given that A AB (ii) BA AB BA , this question illustrates that matrix multiplication is not commutative. (i) Note: Summary: 1. Matrix multiplication is associative. 2. Matrix multiplication is distributive over addition and subtraction. 3. Matrix multiplication is not commutative. III Word Problems 1. An ice-cream stall sells both green tea and mocha ice cream. A small portion of either costs $0.75 and a large portion costs $1.25. During a short period of time, the number of ice creams sold is own in the table below. small large Green Tea 3 4 Mocha 6 3 (i) Write down a column matrix N, representing the cost of each portion of ice cream. 3 4 , evaluate MN. 6 3 (ii) Given that M (iii) Explain what the numbers given in your answer in (ii) signify. Answers: (i) 0.75 N 1.25 3 4 0.75 6 3 1.25 (ii) MN 7.25 8.25 (iii) $7.25 is the amount received from the sales of green tea ice cream while $8.25 is the amount received from the sale of mocha ice cream. As we can see from the example above, matrices can be used in solving word problems. SBGE/MA/SEC1/BEN/MAT/WS3 2 Practice The matrix below shows the results of John and Paul in a multiple choice test. Correct No attempt 18 20 7 John 0 Paul P= Incorrect The method to award marks is given by: Marks 2 Correct Q = 0 No attempt 1 Incorrect (i) Find the matrix product PQ. (ii) Explain clearly what your answer to (a) represents. Answers Part I 1. columns, rows 2. rows, columns 5 3 11 2 ; BA 1 12.5 3.5 4.5 7 10 10 ; BA( NA) 4 6.5 7 Practice 1(i) AB 1(ii) AB 1 6 2 5 1(iii) AB( NA); BA 2 4 4 4 3 22 6 18 5 12 8 20 Part II 1. A( BC ) ( AB)C 4 2 3 4 2. A( B C ) AB AC 10 3 ( B C ) A BA CA 10 2 7 5 8 4 ; BA 0 0 2 1 3. AB Part III 31 1(ii) John’s score is 31 and Paul’s score is 30. 30 Practice 1(i) PQ SBGE/MA/SEC1/BEN/MAT/WS3 3 Matrices Assignment 3 Do all work on Microsoft Word. Save your work as MatA3_ClassIndex (e.g. MatA3_3M01) Send your completed work to (email address) by (date, time) 1. Evaluate the following: 2 (i) 2 3 1 1 4 2. 3. 2 3 (ii) 2 3 1 2 2 3 2 3 0 (iii) 0 2 1 1 3 1 3 c 2 1 0 9 Given that 1 , find all possible values of c and d. 0 2 d d 11 The charges for hiring a car from three different companies, based on the number of days for which a car is hired and/or the number of kilometres for which the car is driven are as follows: Company A charges $66 per day. Company B charges 48 cents on per kilometre driven. Company C charges $30 per day and 25 cents on per kilometre driven. John needs to hire a car for 4 days to drive 560 kilometres. (i) (ii) 4. Write down two matrices only such that the elements of their product under matrix multiplication give the charge of hiring a car from the three different companies. Find this product. Hence state the company that John should hire the car from in order to save cost. A business makes toy buses and toy lorries. The following table is used in calculating the cost of manufacturing each toy. Bus Lorry Labour (Hours) 6 3 Wood (Blocks) 4 4 Paint (Tins) 3 2 Labour costs $8 per hour, wood costs $1 per block and paint costs $2 per tin. 8 6 4 3 It is given that A , B 1 and C AB . 3 4 2 2 (a) (i) Evaluate C. (ii) Explain what the numbers in your answer represent. (b) In addition, D 100 200 . (i) Evaluate DC. (ii) Explain what your answer represents. (O Level/J98/I/16) SBGE/MA/SEC1/BEN/MAT/WS3 4