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Sec 3 Integrated Mathematics : Matrices Name : ______________________ ( ) Class ( ) Date : ___________ Definition A matrix is a rectangular array of numbers and the numbers are known as the elements of the matrix.. Matrices are convenient ways of summarising numerical data. 2 3 e.g. matrix A = then 2, 3, 8 and 1 are called the elements of matrix A. 8 1 Example 1 A company produced 4 types of central heating radiator, known as types A, B, C and D.A builder buys radiators for all the houses on a new estate. There are 20 small houses, 30 medium-sized houses and 15 large houses. A small house needs 3 radiators of type A, 2 of type B and 2 of type C. A medium-sized house needs 2 radiators of type A, 3 of type C and 3 of type D. A large house needs 1 radiators of type B, 6 of type C and 3 of type D. GCE “O” N02 3 2 2 0 We can represent the above information in the form of a matrix such as 2 0 3 3 . 0 1 6 3 (A) Order of a Matrix Before we introduce the 3 operations (addition, subtraction & multiplication of matrices) you must understand the concept of “the order of a matrix”. The order of a matrix is defined as = number of rows number of columns. The order of each of the following matrices are as follow. 6 2 8 31 1 2 3 4 1 0 2 3 4 0 22 23 5 0 3 13 8 11 (B) Addition and Subtraction of Matrices Only matrices of the same order can be added or subtracted. We simply add or minus the corresponding elements. The following illustrate the addition & subtraction of matrices of order 2 2. a b 1 2 a 1 b 2 c d 3 4 c 3 d 4 a b 1 2 a 1 b 2 c d 3 4 c 3 d 4 (C) Scalar Multiplication of Matrices (Multiplication of a scalar and a matrix) a b 8a 8b 8 c d 8c 8d Notice that in addition, subtraction and scalar multiplication, the order of the matrices is preserved. Example 2 Express the following as a single matrix, 2 1 (a) 2 5 3 0 , (b) 2 1 3 5 3 0 , 0 2 0 5 1 4 2 (c) 3 . 1 2 2 6 10 1 2 14 ANSWERS (a ) 10 (b) 13 6 (c) 6 11 6 (D) Multiplication of Two Matrices To multiply two matrices, the orders of the two matrices NEED NOT be the same. However … given that the orders of two matrices A and B are p q and r s respectively, the product of these two matrices is defined only when q r and the order of the resultant product will be p s . E.g. 5 matrix will result in a 1 5 matrix. 4 matrix multiply by a 4 3 matrix will result in a 2 3 matrix. 2 matrix multiply by a 2 6 matrix will result in a 4 6 matrix. k matrix multiply by a k p matrix will result in a h p matrix The product of a 3 2 matrix and a 3 4 matrix is undefined. d 2 e.g. 1 2 3 4 28 a b c e ad be cf f 6 2 4 h 1 3 matrix multiply by a 3 a ad ae af b d e f bd be bf c cd ce cf a b e f ae bg af bh c d g h ce dg cf dh 1 2 4 6 e.g. 2 2 4 6 4 8 12 3 6 12 18 1 2 2 4 14 20 e.g. 3 4 6 8 30 44 Example 3 State which of the following products are not defined and evaluate the others. 1 1 1 2 0 1 (a) (b) 2 5 0 1 , (c) 2 0 4 3 , , 3 0 3 4 3 0 (d) 1 2 3 5 0 1 , 1 0 2 (e) 0 2 , 3 1 4 0 2 (f) (1 3 2) 3 1 , 5 0 1 2 3 (g) , 3 1 1 1 3 2 1 (h) 2 1 0 0 0 1 2 2 3 (i) 3 4 5 0 5 61 7 2 5 0 1 1 7 (i ) undefined 6 9 ( d ) undefined ( e ) ANS : (a ) ( b ) 10 0 2 ( c ) 2 8 ( f ) 19 5 ( g ) 8 ( h) 1 0 3 10 15 0 3 Example 4 A toothpaste firm supplies tubes of toothpaste to 5 different stores. The number of tubes of toothpaste supplied per delivery to each store, the sizes and sale prices of the tubes, together with the number of deliveries made to each store over a 3-month period are shown in the table below. (i) Write down two matrices only such that the elements of their product under matrix multiplication would give the volume of toothpaste supplied to each store per delivery. (ii) Write down two matrices such that the elements of their product under matrix multiplication would give the number of tubes of toothpaste of each size supplied by the firm over the 3-month period. Find this product. (iii) Using the matrix product found in part (ii) and a further matrix, find the total amount of money which would be obtained from the sale of all the tubes of toothpaste delivered over the 3-month period. GCE”O” June 2002 ANS : 400 300 400 0 600 50 0 (i ) 400 0 600 75 500 300 0 100 600 600 400 400 300 400 0 600 0 (ii ) 13 7 10 5 8 400 0 600 16500 10200 18600 500 300 0 600 600 400 2.10 (iii) 16500 10200 18600 3.00 13500 . Therefore total sales - $13500. 3.75 Example 5 A small manufacturing firm produces four types of product, A, B, C and D. Each product requires three processes–assembly, finishing and packaging. The number of minutes required for each type of product for each process and the cost, in $ per minute, of each process are given in the following table. The firm receives an order for 40 of type A, 50 of type B, 50 of type C and 60 of type D. Write down three matrices such that matrix multiplication will give the total cost of meeting this order. Hence evaluate this total cost. GCE”O” Nov 2003 40 8 6 6 5 50 ANS : 0.60 0.20 0.50 5 4 3 2 =………………….. 1111 . Therefore total cost is $1111.00 50 3 3 2 2 60 Example 6 Given that A, B and C are matrices. State whether the following are true or false. (a) AB + AC = A(B + C) ? (b) AB = BA ? (c) A(BC) = (AB)C ? Ans : True, False, True Warning : Must remember that, in general, if A and B are matrices, AB BA , i.e. matrices are not commutative. In Arithmetic, multiplication of numbers are commutative e.g. 2 3 3 2 . (E) Identity Matrices Before learning how to apply matrices to solve simultaneous equations we must learn the concept of inverse matrices and identity matrices. An identity matrix is a square matrix with the digit 1 as the elements along its main diagonal ( sloping downwards from left to right ) and all the other elements being zeros. 1 0 0 0 1 0 0 1 0 0 1 0 0 E.g etc. , 0 1 0, 0 0 1 0 0 1 0 0 1 0 0 0 1 The identity matrix is usually denoted by a capital letter I. Why identity matrix is called identity matrix? Try multiplying any matrix by an identity matrix. An identity matrix acts something like the number “1” in Arithmetic in that it would not change the identity of a matrix when multiplied by it. 1 2 1 0 1 2 e.g. , 3 4 0 1 3 4 1 5 1 0 1 0 1 5 1 5 8 7 0 1 0 1 8 7 8 7 In other words, for any matrix A , AI = IA = A . (must understand and remember this!) Given that A and B are matrices. State whether the following statement is true or false. “If AB = BA A then B must be an identity matrix.” (F) Determinant of a Matrix (Do not confuse with DISCRIMINANT b 2 4ac ) a b The determinant of a matrix M is defined as the value of ad bc and c d a b ad bc . we write det M c d 2 3 2 3 2 5 3 4 22 . e.g A , then det A 4 5 4 5 (G) Inverse Matrix The inverse matrix of a matrix A is denoted by A 1 . 1 a b 1 d b 1 d b 1 a b If A , then A det A c a ad bc c a c d c d 1 2 5 2 5 4 5 1 4 5 1 e.g the inverse matrix of is given by (2)(4) (5)(3) 3 2 7 3 2 3 4 3 4 Note : (1) A matrix whose determinant is zero has no inverse and hence is known as a singular matrix. (2) Only square matrices have inverses. Warning : Say A, B and C are matrices such that AB = C . And say we know matrices A and C and are asked to find matrix B. We cannot hope to find B by dividing both sides by matrix A. Do you notice, so far, we have not introduce “Division of Matrices”. Reason : Such operation is not defined in this branch of mathematics calls matrices. Then how can we find matrix B? This is where the inverse matrix concept comes in. You see to find B, we need to eliminate matrix A from the left side of B. How can we eliminate A? Well, by multiplying by the inverse of A… AB = C A1AB = A1C (NOT AA1B = CA1!) You must multiply matrix A by its inverse A 1 on the left hand side of A ( not right hand side! ) But A 1A = I hence we can simplify to IB = A1C B = A1C Bingo! Example 7 2 1 4 2 Given that A = and C . Find the matrix B such that AB = C . 2 3 3 1 ANS : From the above we get B = A1C Now we have A1 3 1 1 3 1 1 , (2)(3) (1)(2) 2 2 8 2 2 1 3 1 4 2 1 9 5 Hence B = A 1C 8 2 2 3 1 8 14 6 (H) Using Matrices to solve simultaneous equation We can solve a pair of simultaneous equations ax by m by expressing it in a matrix form as cx dy n a b x m c d y n 1 1 a b a b x a b m c d c d y c d n 1 1 0 x a b m 0 1 y c d n 1 x a b m y c d n Example 8 (a) Use the inverse matrix method to solve the simultaneous equations 6 x 3 y 15 and 4 x 2 y 2 . What do you notice? 6 3 x 15 Ans : Converting into matrix form, we get . 4 2 y 2 6 3 But det is equal to zero, 6 2 3 4 0 4 2 Indeed 6 x 3 y 15 y 2 x 5 , 4 x 2 y 2 y 2 x 1 , they are equations of parallel lines. Since there is no point of intersection between the two lines, there is no solution. Example 8 (b) Use the inverse matrix method to solve the simultaneous equations 9 x 6 y 6 0 and 3x 2 y 2 0 . What do you notice? 9 6 x 6 Ans : Converting into matrix form, we get . 3 2 y 2 9 6 But det is also equal to zero, 9 2 6 3 0 3 2 Both equations could be simplified to y 3 x 1 . They give rise to identical graph. Therefore there are 2 infinite number of solutions. Example 9 2005 2006 1 1 2 as a single matrix. Given that B , write down BB 2007 2008 3 4 Ans : Do not spend time computing BB 1 , it is equal to identity matrix, hence 1 2 1 0 1 2 1 2 1 2 BB1 I . 3 4 0 1 3 4 3 4 3 4 End of notes Sec 3 Integrated Mathematics Enrichment Worksheet Topic : Application of Matrices to Cryptography Name : ______________________ ( ) Class ( ) Date : ___________ (1) To encode message using matrices, we must first assign numbers to letters of the alphabet. Let A = 1, B = 2, etc as shown in the table below. A=1 B=2 C=3 D=4 Space = 27 E=5 F=6 G=7 H=8 I=9 J = 10 K = 11 L = 12 M = 13 N = 14 O = 15 P = 16 Q = 17 R = 18 S = 19 T = 20 U = 21 V = 22 W = 23 X = 24 Y = 25 Z = 26 (2) Say the message you want to encode is “HWA CHONG ”. You need to break the message into groups of 2 letters (and spaces) each. The message will become HW A_ CH ON G_ (3) Convert each block of 2 letters into a 2 1 matrix. You will get the following sequence of matrices 8 1 3 15 7 23 27 8 14 27 2 0 (4) Now use any 2 2 “multiplier” matrix that has an inverse. Say we choose M = . 0 1 Find the product of M and each of the preceding matrices gives 16 2 6 30 14 23 27 8 14 27 (5) The recipient of will receive the encoded message as a string of numbers …. 16 23 2 27 6 8 30 14 14 27 which he has to convert them back to sequence of 2 1 matrices. After multiplying by the inverse, M 1 , the message will be revealed. Challenge Yourself You received the following encoded message. 26 21 21 13 31 18 37 19 51 39 You have earlier agreed with the encoder that the multiplier matrix is What is the encoded message? 2 1 . 1 1