7Matrices and ONLINE PAGE PROOFS. applications to transformations

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1 7Matrices and applications to transformations 7. Kick off with CAS 7. Addition, sutraction and scalar multiplication of matrices 7. Matri multiplication 7.4 Determinants and inverses of matrices 7.5 Matri equations and solving linear simultaneous equations 7.6 Translations 7.7 Reflections 7.8 Dilations 7.9 Cominations of transformations 7.0 Review

2 7. Kick off with CAS Matrices Using CAS technolog, create the following matrices. 6 a 5 4 c c d 8 6 Using CAS technolog, calculate each of the following a c c 5 d What do ou notice aout our answers to question? 4 Using CAS technolog, define the following matrices. A = 4, B = 5 0, and I = Calculate each of the following. a 5A B c A + B d det A e B f BI 6 Eamine the answers to question 5. What do ou notice? Please refer to the Resources ta in the Prelims section of our ebookplus for a comprehensive step--step guide on how to use our CAS technolog.

3 Units & AOS Topic Concept 7. Addition, sutraction and scalar multiplication of matrices Concept summar Practice questions Addition, sutraction and scalar multiplication of matrices Introduction to matrices The tale shows the final medal tall for the top four countries at the 0 London Olmpic Games. Countr Gold Silver Bronze United States of America People s Repulic of China 8 7 Great Britain Russian Federation 4 5 This information can e presented in a matri, without the countr names, and without the headings for gold, silver and ronze: The data is presented in a rectangular arra and is called a matri. It conves information such as that the second countr won 8 gold, 7 silver and ronze medals. This matri has four rows and three columns. The numers in the matri, in this case representing the numer of medals won, are called elements of the matri. Matrices In general we enclose a matri in square rackets and usuall use capital letters to denote it. The size or order of a matri is important, and is determined the numer of rows and the numer of columns, strictl in that order. Consider the following set of matrices: A = A is a matri: it has two rows and two columns. B = B is a matri: it has three rows and three columns. When the numers of rows and columns in a matri are equal, it is called a square matri. C = 50 Maths Quest MATHEMATICAL METHODS VCE Units and

4 C is a matri as it has three rows and one column. If a matri has onl one column, it is also called a column or vector matri. D = D is a matri as it has one row and two columns. If a matri has onl one row it is also called a row matri. E = 5 4 E is a matri as it has three rows and two columns. F = F is a matri as it has two rows and three columns. Each element in a matri can also e identified its position in the matri. We use suscripts to identif the row and column numer. For eample, in matri A = 5 4 7, the element is in the first row and first column, so a =. The element 5 is in the first row and second column, so a = 5. The element 4 is in the second row and first column, so a = 4. Finall, the element 7 is in the second row and second column, so a = 7. In general, we can write a matri as: A = a a a a A general matri of order m n can e written as: a a a a n a a a... a n a a a... a n a m a m a m... a mn Here, a 4 denotes the element in the first row and fourth column, a 4 denotes the element in the fourth row and third column, and a ij denotes the element in the ith row and jth column. Operations on matrices Operations include addition, sutraction and multiplication of two matrices. Note that we cannot divide matrices. Equalit of matrices Two matrices are equal if and onl if the have the same size or order and each of the corresponding elements are equal. Topic 7 Matrices and applications to transformations 5

5 For eample, if z = then a =, = 5, c = 4 and d = 7. Addition of matrices, then =, = and z = ; if a c d = 5 4 7, Onl two matrices of the same size or order can e added together. To add two matrices, we add the elements in the corresponding positions. For eample, if 4 P = and Q = 5 6, then: P + Q = = = If two matrices cannot e added together (if the sum does not eist), we sa that the two matrices are not conformale for addition. Scalar multiplication of matrices To multipl an matri (of an size or order) a scalar, we multipl ever element in the matri the scalar. If P = 5 then: P = 5 = = 0 Sutraction of matrices Onl two matrices of the same size or order can e sutracted. To sutract two matrices, we sutract the elements in the corresponding positions. For eample, if 4 P = and Q = 5 6, then: P Q = P + Q = = = 5 7 If two matrices cannot e sutracted (if the difference does not eist), we sa that the matrices are not conformale for sutraction. Note that none of the matrices defined A = , B = 4 5, C =, D =, E = 4, 4 5 Maths Quest MATHEMATICAL METHODS VCE Units and

6 F = different orders. can e added or sutracted from one another, as the are all of WOrKeD example think At a footall match one food outlet sold 80 pies, 0 hotdogs and 0 oes of chips. Another food outlet sold 00 pies, 0 hotdogs and 90 oes of chips. Represent this data as a matri, and find the total numer of pies, hotdogs and chips sold these two outlets. Use a matri to represent the numer of pies, hotdogs and chips sold. Write down the matri for the sales from the first outlet. Write down the matri for the sales from the second outlet. WritE pies hotdogs chips S = S = Find the sum of these two matrices. S + S = = WOrKeD example think Given the matrices A = and if A + B = C. 5, B = 5 4 and C = WritE Sustitute for the given matrices. A + B = C = Appl the rules for scalar multiplication. Appl the rules for addition of matrices = = 4 Appl the rules for equalit of matrices. 0 = = 5 Solve the first equation for. = = 4 6 Sustitute for into the second equation and solve this equation for = = 8 0 = find the values of topic 7 MatrICes and applications to transformations 5

7 WOrKeD example think special matrices the zero matri The null matri or zero matri O, with all elements equal to zero, is given O = When matrices A + B = O, matri B is the additive inverse of A. So, B = O A. the identit matri The identit matri I is defined I = 0. This matri has ones down the 0 leading diagonal and zeros on the other diagonal. Given the matrices A = and B = 4, find the matri X if: a X = A B A + X = O c X = B + A I a Sustitute for the given matrices. WritE a X = A B = Appl the rules for scalar multiplication. = 6 0 Appl the rules for sutraction of matrices. = Transpose the equation to make X the suject. State the final answer. = c Sustitute for the given matrices. Appl the rules for scalar multiplication. = Appl the rules for addition and sutraction of matrices A + X = O X = O A = c X = B + A I = = State the final answer. = Maths Quest MatheMatICaL MethODs VCe units and

8 Eercise 7. PRactise Work without CAS Consolidate Appl the most appropriate mathematical processes and tools Addition, sutraction and scalar multiplication of matrices WE At footall matches, commentators often quote plaer statistics. In one particular game, the top ranked plaer on the ground had 5 kicks, 8 marks and 0 handalls. The second ranked plaer on the same team on the ground had 0 kicks, 6 marks and 8 handalls, while the third ranked plaer on the same team on the ground had 8 kicks, 5 marks and 7 handalls. Represent this data as a matri, and find the total numer of kicks, marks and handalls these three plaers from the same team. At the end of a doules tennis match, one plaer had aces, doule faults, 5 forehand winners and 0 ackhand winners, while his partner had 4 aces, 5 doule faults, 8 forehand winners and 7 ackhand winners. Represent this data as a matri and find the total numer of aces, doule faults, forehand and ackhand winners for these plaers. WE Given the matrices A = 4, B = 4 5 and C = and if A + B = C. 4 Given the matrices A = 4, B = 5 and z if A B = C. 5 WE If A = 4 5 and B = 4 and C = 6 z find the values of find the values of, find matri X given the following. a X = A B A + X = O c X = B A + I 6 If A = a c d and B = 4 a A + I B = O 7 If A = and B = 5 find the values of a,, c and d given the following. I + 4B A = O find matri C given the following. a C = A + B A + C = B c A + C = 4B 8 If A = and B = 5 find matri C given the following. a C = A + B A + C = B c A + C = 4B 9 Consider these matrices: A = 4, B = 4 5 and C = 5 4 a Find the following matrices. i B + C ii A + B Verif the Associative Law for matri addition: A + (B + C) = (A + B) + C. 0 If A = 4, B = 4 5 and O = find matri C given the following. a A = C B C + A B = O c C + A B = O Topic 7 Matrices and applications to transformations 55

9 4 Given the matrices A =, B = matri C if the following appl. a A + C B + 4I = O 4 5, O = and I = 0 0, find 4A C + B I = O If A = and B = find the values of and given the following. Master a A + B = If D = 4 5 and E = A B = c B A = 4 6 find matri C given the following. a C = D + E D + C = E c D + C = 4E 4 If D = and E = 4 find the matri C given the following. a C = D + E D + C = E c D + C = 4E Given A = 4, answer questions 5 and 6. 5 a Write down the values of a, a, a and a. Find the matri A if a =, a =, a = and a = 5. 6 a Find the matri A whose elements are a i j = i j for j i and a i j = i j for j = i. Find the matri A whose elements are a i j = i + j for i < j, a i j = i j + for i > j and a i j = i + j + for i = j. 7 The trace of a matri A denoted tr(a) is equal to the sum of leading diagonal elements. For matrices, if A = a a a a then tr(a) = a + a. Consider the following matrices: A = 4, B = 4 and C = a Find the following. i tr(a) ii tr(b) iii tr(c) Is tr(a + B + C) = tr(a) + tr(b) + tr(c)? c Is tr(a + B 4C) = tr(a) + tr(b) 4tr(C)? 8 If A = and B = 4 0 What tpe of matri is AB? 5 4, use our calculator to calculate A B. 56 Maths Quest MATHEMATICAL METHODS VCE Units and

10 7. Units & AOS Topic Concept Matri multiplication Concept summar Practice questions Matri multiplication Multipling matrices At the end of an AFL footall match etween Sdne and Melourne the scores were as shown. This information is represented in a matri as: Goals Sdne 5 Melourne 9 0 Behinds One goal in AFL footall is worth 6 points and one ehind is worth point. This information is represented in a matri as: Goals Behinds To get the total points scored oth teams the matrices are multiplied = = This is an eample of multipling a matri a matri to otain a matri. Multipling matrices in general Two matrices A and B ma e multiplied together to form the product AB when the numer of columns in A is equal to the numer of rows in B. Such matrices are said to e conformale with respect to multiplication. If A is of order m n and B is of order n p, then the product has order m p. The numer of columns in the first matri must e equal to the numer of rows in the second matri. The product is otained multipling each element in each row of the first matri the corresponding elements of each column in the second matri. In general, if A = a c d and B = d ad + e then AB = e cd + de. For matrices if A = a a a a and B = then: In general AB BA. AB = a + a a + a a + a a + a Topic 7 Matrices and applications to transformations 57

11 WOrKeD example 4 Given the matrices A = and X = a AX XA, find the following matrices. think WritE a Sustitute for the given matrices. a AX = Appl the rules for matri multiplication. Since A is a matri and X is a matri, the product AX will e a matri. WOrKeD example 5 think Given the matrices A = and B = 4 a AB BA c B a Sustitute for the given matrices. Since A and B are oth matrices, the product AB will also e a matri. WritE a AB = Appl the rules for matri multiplication. = Simplif and give the final result. = 0 6 Sustitute for the given matrices. Since oth A and B are matrices, the product BA will also e a matri. BA = Appl the rules for matri multiplication. = Simplif and give the final result. = 5 4 find the following matrices AX = Simplif and give the final result. AX = Appl the rules for matri multiplication. Since X is a matri and A is a matri, the product XA does not eist ecause the numer of columns of the first matri is not equal to the numer or rows in the second matri XA does not eist. c B = B B. Write the matrices. c B = 5 4 = Maths Quest MatheMatICaL MethODs VCe units and

12 Since B is a matri, B will also e a matri. Appl the rules for matri multiplication. Simplif and give the final result. = 6 7 = WOrKeD example 6 think a Sustitute for the given matrices. These last two eamples show that matri multiplication in general is not commutative: AB BA, although there are eceptions. It is also possile that one product eists and the other simpl does not eist, and that the products ma have different orders. Note that squaring a matri (when defined) is not the square of each individual element. 5 Given the matrices E = 4 matrices. a EF FE Appl the rules for matri multiplication. Since E is a matri and F is a matri, the product EF will e a matri. Simplif and give the final result. Sustitute for the given matrices. Appl the rules for matri multiplication. Since F is a matri and E is a matri, the product EF will e a matri. Simplif and give the final result. WritE a EF = = = 5 4 and F = , find the following FE = = = topic 7 MatrICes and applications to transformations 59

13 Eercise 7. PRactise Work without CAS Consolidate Appl the most appropriate mathematical processes and tools Matri multiplication WE4 Given the matrices A = 4 5 and X =, find the following matrices. a AX Given the matrices A = a c d and X = a AX XA find the following matrices. XA WE5 Given the matrices A = 4 5 and B = 4 following matrices. find the a AB BA c A 4 Given the matrices A = 4 5, O = and I = 0 0 following matrices. find the a AO OA c AI d IA What do ou oserve from this eample? 5 WE6 Given the matrices D = following matrices. 5 4 and E = 4 5 find the a DE ED 6 Given the matrices C = and D = find the following matrices. a CD DC 7 a Given the matrices A = 4, I = 0 and O = following. verif the i AI = IA = A ii AO = OA = O Given the matrices A = a c d I = 0 0 and O = 0 0 verif the 0 0 following. i AI = IA = A ii AO = OA = O 8 a Given the matrices A = 4 and I = 0 0 (I A) (I + A) = I A. 0 If A = 4 0 and B = 0 0 Does BA = O? c If A = a 0 0 and B = 0 0 Does BA = O? verif that show that AB = O where O = show that AB = O where O = Maths Quest MATHEMATICAL METHODS VCE Units and

14 9 Consider the matrices A = 4, B = 4 5 and C = 5 4. a Verif the Distriutive Law: A(B + C) = AB + AC. Verif the Associative Law for Multiplication: A(BC) = (AB)C. c Is (A + B) = A + AB + B? Eplain. d Show that (A + B) = A + AB + BA + B. 0 If A = and B = a AB = 8 c A = 8 find the value of given the following. BA = d B = Given the matrices A =, B = 5 and C = 4 each of the following matrices. 5 find, if possile, a A + B A + C c B + C d AB e BA f AC g CA h BC i CB j ABC k CBA l CAB Given D = 4 and E = find the following matrices. a DE ED c E + D d D Given D = matrices and E = 6 find the following a DE ED c E + D d D 4 a If P = find the matrices P, P, P 4 and deduce the matri P n. If Q = 0 0 find the matrices Q, Q, Q 4 and deduce the matri Q n. c If R = 0 find the matrices R, R, R 4 and deduce the matri R n. d If S = 0 0 find the matrices S, S, S 4, and deduce the matrices S 8 and S 9. 5 If A = 6 a If B = If C = 4 and I = 0 0 evaluate the matri A 6A + I. 4 5 and I = 0 0 evaluate the matri B B I. 5 4 and I = 0 0 evaluate the matri C 5C + 4I. Topic 7 Matrices and applications to transformations 6

15 MAstEr 7.4 Units & AOS Topic Concept Determinants and inverses of matrices Concept summar Practice questions WOrKeD example 7 think 7 If D = d 4 8 evaluate the matri D 9D. 8 The trace of a matri A denoted tr(a) is equal to the sum of leading diagonal elements. For matrices if A = a a a a then tr(a) = a + a. Consider the matrices A = 4 B = 4 5 and C = 5 4. a Find the following. i tr(ab) ii tr(ba) iii tr(a)tr(b) Is tr(abc) = tr(a)tr(b)tr(c)? Determinants and inverses of matrices Determinant of a matri Associated with a square matri is a single numer called the determinant of a matri. For matrices, if A = a then the determinant of the matri A is c d denoted det(a) or often given the smol Δ. The determinant is represented not the square rackets that we use for matrices, ut straight lines; that is, det(a) = Δ = A = a. To evaluate the determinant, multipl the elements c d in the leading diagonal and sutract the product of the elements in the other diagonal: det(a) = Δ = A = a = ad c. c d Find the determinant of the matri F = WritE Appl the definition and multipl the elements in F = 5 the leading diagonal. Sutract the product of the 4 7 elements in the other diagonal. det(f) = = 0 State the value of the determinant. det(f) = Inverses of matrices The multiplicative identit matri I, defined I = 0, has the propert that 0 for a non-zero matri A, AI = IA = A. When an square matri is multiplied its multiplicative inverse, the identit matri I is otained. The multiplicative inverse is called the inverse matri and 6 Maths Quest MatheMatICaL MethODs VCe units and

16 WOrKeD example 8 think is denoted A. Note that A as division of matrices is not defined; A furthermore, A A = A A = I. Consider the products of A = and 7 5 A = 4 : A A = A A = Now for the matri A = = = = = the determinant 4 7 =. 7 5 A = is otained from the matri A swapping the elements on the 4 leading diagonal, and placing a negative sign on the other two elements. These results are true in general for matrices, ut we must also account for the value of a non-unit determinant. To find the inverse of a matri, the value of the determinant is calculated first; then, provided that the determinant is non-zero, we divide the determinant, then swap the elements on the leading diagonal and place a negative sign on the other two elements. In general if A = a c d then the inverse matri A is given A d = ad c c a. We can show that A A = A A = I. Find the inverse of the matri P = Calculate the determinant. If P = a, then P = ad c. c d To find the inverse of matri P, appl the rule P d = ad c c a. WritE Sustitute and evaluate P P. P P = 5 and verif that P P = P P = I. P = 5 = 5 = 0 + = P = 5 4 Appl the rules for scalar multiplication and multiplication of matrices. = topic 7 MatrICes and applications to transformations 6

17 5 Simplif the matri product to show that P P = I. WOrKeD example 9 think Evaluate the determinant. singular matrices If a matri has a zero determinant then the inverse matri simpl does not eist, and the original matri is termed a singular matri. Show that the matri 6 4 Since the determinant is zero, the matri is singular. 6 4 WOrKeD example 0 think WritE 6 4 is singular. 6 4 = 0 = ( 4) (6 ) = = 0 If A = 4 5 and I = 0, epress the determinant of the matri A ki 0 in the form pk + qk + r and evaluate the matri pa + qa + ri. Sustitute to find the matri A ki. Appl the rules for scalar multiplication and sutraction of matrices. Evaluate the determinant of the matri A ki. Simplif the determinant of the matri A ki. WritE = 6 Sustitute and evaluate P P. P P = 7 Use the rules for scalar multiplication and multiplication of matrices. 8 Simplif the matri product to show that P P = I. = = 0 0 = = 0 0 A ki = 4 5 k 0 0 = k 4 5 k det A ki = k 4 5 k = ( k)(5 k) 4 = ( + k)(5 k) = (0 + k k ) = k k 64 Maths Quest MatheMatICaL MethODs VCe units and

18 4 State the values of p, q and r. p = ; q = ; r = 5 Determine the matri A. A = 4 6 Sustitute for p, q and r and evaluate the matri A A I. 7 Simplif appling the rules for scalar multiplication of matrices. 8 Simplif and appl the rules for addition and sutraction of matrices. Eercise 7.4 PRactise Work without CAS Consolidate Appl the most appropriate mathematical processes and tools Determinants and inverses of matrices WE7 Find the determinant of the matri G = 4 5. The matri 5 + has a determinant equal to 9. Find the values of. WE8 Find the inverse of the matri A = and verif that A A = A A = I. 4 The inverse of the matri 4 is p. Find the values of p and q. q 5 WE9 Show that the matri is singular. 6 Find the value of if the matri 4 is singular WE0 If A = 5 and I = 0 epress the determinant of the matri A ki 0 in the form pk + qk + r and evaluate the matri pa + qa + ri If A = and I = 0 find the value of k for which the determinant of 0 the matri A ki is equal to zero. 5 A A I = = Consider the matri P = 6 4. a Find the following. i det(p) ii P Verif that P P = P P = I. c Find the following. i det(p ) ii det(p) det(p ) 0 Find the inverse matri of each of the following matrices. 0 0 a c = A A I = d 0 Topic 7 Matrices and applications to transformations 65

19 Master The following matrices refer to questions and. A = 4, B = 4 5 and C = 4 a Find det(a), det(b) and det(c). Is det(ab) = det(a)det(b)? c Verif that det(abc) = det(a)det(b)det(c). a Find the matrices A, B, C. Is (AB) = A B? c Is (AB) = B A? d Is (ABC) = C B A? Find the value of for each of the following. a 4 = 6 8 = c 4 = 4 d = 7 4 Find the values of if each of the following are singular matrices. a + c d Given A =, B = 5 and C = 4 find, if possile, the 5 following matrices. a (AB) A c B d C e (ABC) 6 If A = 4 and I = 0 epress the determinant of the matri A ki, 0 k R in the form pk + qk + r, and evaluate the matri pa + qa + ri. 7 If B = 4 5 and I = 0 epress the determinant of the matri B ki, k R 0 in the form pk + qk + r, and evaluate the matri pb + qb + ri. 8 Consider the matrices A =, P = and I = 0 0. a Find the values of k, for which the determinant of the matri A ki = 0. Find the matri P AP. 9 Consider the matrices B = 5 4, Q = 5 and I = 0 0. a Find the values of k for which the determinant of the matri B ki = 0. Find the matri Q BQ. cos θ sin θ 0 Let R θ =. sin θ cos θ a Use a CAS technolog, or otherwise, to find the following matrices. i R π ii R π 6 Show that R π 6 R π = R π. c Show that R(α)R(β) = R(α + β). iii R π iv R CAS v R 66 Maths Quest MATHEMATICAL METHODS VCE Units and

20 7.5 Units & AOS Topic Concept 4 Matri equations and solving linear simultaneous equations Concept summar Practice questions WOrKeD example think Matri equations and solving linear simultaneous equations sstems of simultaneous linear equations Consider the two linear equations a + = e and c + d = f. These equations written in matri form are: a c d = e f This is the matri equation AX = K, where A = a c d, X = and K = e f. Recall that A = d where Δ = ad c, and that this matri has the Δ c a propert that A A = I = 0 0. To solve the matri equation AX = K, pre-multipl oth sides of the equation AX = K A. Recall that the order of multipling matrices is important. A AX = A K, since A A = I IX = A K X = A K X = = ad c Solve for and using inverse matrices = 6 + = 8 First rewrite the two equations as a matri equation. WritE 4 5 = 6 8 and IX = X d c a Write down the matrices A, X and K. A = 4 5, X = and K = 6 8 Find the determinant of the matri A. Δ = 4 5 = 4 5 = 7 e f 4 Find the inverse matri A, and appl the rules for scalar multiplication to simplif this inverse. A = = topic 7 MatrICes and applications to transformations 67

21 5 The unknown matri X satisfies the equation X = A K. Write the equation in matri form. X = = Appl the rules for matri multiplication. The product is a matri. = = Appl the rules for scalar multiplication, and the rules for equalit of matrices. WOrKeD example think Solve the following simultaneous linear equations for and. = = 0 First write the two equations as matri equations. WritE Write down the matrices A, X and K. A = 6 4 Find the determinant of the matri A. Δ = The inverse matri A does not eist. This method cannot e used to solve the simultaneous equations. 5 Appl the method of elimination first numering each equation. 6 To eliminate, multipl equation () and add this to equation (). 7 In eliminating, we have also eliminated and otained a contradiction. = = 7 8 State the final answer. = 4 and = = , X = and K = 6 0 The matri A is singular. = = 0 = ( 4) ( 6 ) = = = 0 0 =? 8 4 = 4 8 Appl another method to solving simultaneous equations: the graphical method. Since oth equations represent straight lines, determine the - and -intercepts. Line = 6 crosses the -ais at (, 0) and the -ais at 0,. Line = 0 crosses the -ais at 5, 0 and the -ais at 0, Maths Quest MatheMatICaL MethODs VCe units and

22 9 Sketch the graphs. Note that the two lines are parallel and therefore have no points of intersection. ( ) 0, = 0 = 6 ( ) 5, 0 (, 0) 0 (0, ) 0 State the final answer. There is no solution. WOrKeD example think Solve the following linear simultaneous equations for and. = = First write the two equations as matri equations. WritE Write down the matrices A, X and K. A = 6 4 Find the determinant of the matri A. Δ = The inverse matri A does not eist. This method cannot e used to solve the simultaneous equations. 5 Appl another method of solving simultaneous equations: elimination. Numer the equations. 6 To eliminate, multipl equation () and add this to equation (). 7 In eliminating, we have also eliminated ; however, we have otained a true consistent equation. = 6 6 4, X = and K = 6 The matri A is singular. = = 6 4 = = 0 = 0 = ( 4) ( 6 ) = 0 8 Appl another method of solving simultaneous equations: the graphical method. Determine the - and -intercepts. = 6 (line ) crosses the -ais at (, 0) and the -ais at 0, = (line ) is actuall the same line, since () = (). topic 7 MatrICes and applications to transformations 69

23 9 Sketch the graphs. Note that since the lines overlap, there is an infinite numer of points of intersection. = = (, 0) 0 (0, ) Since = 6 = 6 + If = 0, = (, 0) If =, = 8 8, If =, = 0 0, If = = 4 (4, ) In general, let = t so that = 6 + t. 6 + t As a coordinate:, t 0 State the final answer. There is an infinite numer of solutions of the form + t, t where t R. Geometrical interpretation of solutions Consider the simultaneous linear equations a + = e and c + d = f and the determinant: det(a) = Δ = A = a = ad c c d If the determinant is non-zero (Δ 0) then these two equations are consistent; graphicall, the two lines have different gradients and therefore the intersect at a unique point. ( ) If the determinant is zero (Δ = 0) then there are two possiilities. The onl certaint is that there is not a unique solution. 0 If the lines are parallel, the equations are inconsistent and there is a contradiction; this indicates that there is no solution. That is, graphicall the two lines have the same gradient ut different -intercepts Maths Quest MATHEMATICAL METHODS VCE Units and

24 If the lines are simpl multiples of one another, the equations are consistent and dependent. This indicates that there is an infinite numer of solutions. That is, the have the same gradient and the same -intercept (the overlap). 0 WOrKeD example 4 Find the values of k for which the equations k = k and 0 (k + ) = 8 have: a a unique solution no solution c an infinite numer of solutions. (You are not required to find the solution set.) think WritE k a First write the two equations as matri equations. a 0 (k + ) Write out the determinant, as it is the ke to Δ = k answering this question. 0 (k + ) = k 8 Evaluate the determinant in terms of k. Δ = k k = k k + 0 = k + k 0 4 If the solution is unique then Δ 0; that is, there Δ = (k + 6) (k 5) is a unique solution when k 6 and k 5. There is a unique solution when k 6 Now investigate these two cases. and k 5, or k R \ 6, 5. 5 Sustitute k = 6 into the two equations. () 6 = 7 + = 7 If there is no solution then the two equations represent parallel lines with different -intercepts. Interpret the answer. () = 8 + = 8 5 When k = 6 there is no solution. Sustitute k = 5 into the two equations. () 5 = 4 () 0 6 = 8 c If there are infinite solutions, the two equations are multiples of one another. Interpret the answer. c When k = 5 there is an infinite numer of solutions. topic 7 MatrICes and applications to transformations 7

25 WOrKeD example 5 Matri equations In matri algera, matrices are generall not commutative. The order in which matrices are multiplied is important. Consider the matri equations AX = B and XA = B, where all the matrices A, B and X are matrices, the matrices A and B are given, and in each case the unknown matri X needs to e found. If AX = B, to find the matri X oth sides of the equation must e pre-multiplied A, the inverse of matri A. A AX = A B since A A = I IX = A B where I = 0 ; recall that this identit matri satisfies IX = X 0 so that if AX = B then X = A B. If XA = B, to find the matri X, post-multipl oth sides of the equation A. XAA = BA since AA = I XI = BA where I = 0 ; recall that this identit matri satisfies XI = X so 0 that if XA = B then X = BA. Given the matrices A = and B = 4 a AX = B XA = B. find matri X, if: think WritE a Evaluate the determinant of the matri A. a det A = 5 = = 4 6 Find the inverse matri A. A = If AX = B, pre-multipl oth sides the X = inverse matri A ; then X = A B X is a matri. Appl the rules for multipling matrices. = 5 State the answer. = If XA = B, post-multipl oth sides the inverse matri A so that X = BA. X is a matri. Appl the rules to multipl the matrices X = = = State the answer. = Maths Quest MatheMatICaL MethODs VCe units and

26 WOrKeD example 6 Given the matrices A = 4 5 6, C = and D = = [ ] find matri X if: a AX = C XA = D. think EErcisE 7.5 PrActisE Work without cas WritE a Evaluate the determinant of the matri A. a det(a) = Find the inverse matri A. A = If AX = C then pre-multipl oth sides AX = C the inverse matri A. A AX = A C 4 Sustitute for the given matrices. IX = X = A C 5 X is a matri. Appl the rules to multipl the matrices. X = = Matri equations and solving linear simultaneous equations WE Solve for and using inverse matrices. 4 = 5 + = Solve for and using inverse matrices. + 5 = 7 = State the answer. = 4 If XA = D, post-multipl oth sides the inverse matri A. XA = D XAA = DA Sustitute for the given matrices. XI = X = DA X is a matri. Appl the rules to multipl the matrices. X = = = = = State the answer. = 4 WE Solve the following linear simultaneous equations for and. 4 = = 8 topic 7 MatrICes and applications to transformations 7

27 Consolidate toolsonline Appl the most appropriate mathematical processes and 4 Find the value of k, if the following linear simultaneous equations for and have no solution. 5 4 = 0 k + = 8 5 WE Solve the following linear simultaneous equations for and. 4 = = 4 6 Find the value of k if the following linear simultaneous equations for and have an infinite numer of solutions. 5 4 = 0 k + = 0 7 WE4 Find the values of k for which the sstem of equations (k + ) = k and 6 + k = 8 has: a a unique solution no solution c an infinite numer of solutions. (You are not required to find the solution set.) 8 Find the values of k for which the sstem of equations + (k ) = 4 and k + 6 = k + 4 has: a a unique solution no solution c an infinite numer of solutions. (You are not required to find the solution set.) 9 WE5 If A = 4 and B = 4 5 find matri X given the following. a AX = B XA = B 0 If P = 4, Q = 6 and O = find matri X given the following. a XP Q = O PX Q = O WE6 If A = 4 5, C = following. a AX = C 5 If B = 4, C = a BX = C and D = 5 find matri X given the XA = D and D = 4 find matri X given the following. XB = D Solve each of the following simultaneous linear equations using inverse matrices. a + = = 9 c = = 6 PAGE PROOFS = 8 d = = = 0 4 Solve each of the following simultaneous linear equations using inverse matrices. a + 4 = 6 + = = 5 = 74 Maths Quest MATHEMATICAL METHODS VCE Units and

28 c 4 + = = 5 d + 5 = 5 = 6 5 a The line a + = passes through points (, 6) and (8, ). i Write down two simultaneous equations that can e used to solve for a and. ii Using inverse matrices, find the values of a and. The line a + = passes through points (4, 5) and ( 4, 5). i Write down two simultaneous equations that can e used to solve for a and. ii Using inverse matrices, find the values of a and. 6 Find the values of k for which the following simultaneous linear equations have: i no solution ii an infinite numer of solutions. a = k + 6 = 6 c 5 = k = 0 5 = = k d 4 6 = 8 + = k 7 Show that each of the following does not have a unique solution. Descrie the solution set and solve if possile. a = + 4 = 6 = = c = = 7 d 4 = = 0 8 Find the values of k for which the sstem of equations has: i a unique solution ii no solution iii an infinite numer of solutions. (You are not required to find the solution set.) a (k ) = k 4 + k = 6 (k + ) + 5 = k = k + 6 c (k ) = k + d (k ) = k = 0 (k 5) = k 9 Find the values of p and q for which the sstem of equations has: i a unique solution ii no solution iii an infinite numer of solutions. (You are not required to find the solution set.) a + = p q 6 = 7 c p = 6 7 = q 4 = q + p = 0 d p = + = q Topic 7 Matrices and applications to transformations 75

29 Master 7.6 Units & AOS Topic Concept 5 Translations Concept summar Practice questions 0 Consider the matrices A = 5 4, B = 7, C = 5 9 Find the matri X in each of the following cases. and D = 7 4. a AX = C XA = B c AX = B d XA = D Consider the matrices A = 4, B = 4 5, C = 5 4 Find the matri X in each of the following cases. and D =. a AX = C XA = B c AX = B d XA = D Consider matrices A = 4 5, B = 9 7, C = the matri X in each of the following cases. and D =. Find a AX = C XA = B c AX = B d XA = D a,, c and d are all non-zero real numers. a If P = a 0 0 d find P and verif that PP = P P = I. If Q = 0 c 0 find Q and verif that QQ = Q Q = I. 4 a,, c and d are all non-zero real numers. a If R = a c 0 find R and verif that RR = R R = I. If S = c If A = 0 c d find S and verif that SS = S S = I. a Translations c d write down A and verif that AA = A A = I. Introduction to transformations Computer engineers perform matri transformations in the computer animation used in movies and video games. The animation models use matrices to descrie the locations of specific points in images. Transformations are added to images to make them look more realistic and interesting. Interactivit Translations int Maths Quest MATHEMATICAL METHODS VCE Units and

30 Matri transformations A transformation is a function which maps the points of a set X, called the pre-image, onto a set of points Y, called the image, or onto itself. A transformation is a change of position of points, lines, curves or shapes in a plane, or a change in shape due to an enlargement or reduction a scale factor. Each point of the plane is transformed or mapped onto another point. The transformation, T, is written as: T : which means T maps the points of the original or the pre-image point (, ) onto a new position point known as the image point (, ). An transformation that can e represented a matri, a, is called a c d linear transformation. The origin never moves under a linear transformation. An invariant point or fied point is a point of the domain of the function which is mapped onto itself after a transformation. The pre-image point is the same as the image point. = An invariant point of a transformation is a point which is unchanged the transformation. For eample, a reflection in the line = leaves ever point on the line = unchanged. The transformations which will e studied in this chapter are translations, reflections, rotations and dilations. When two or more transformations are applied to a function, the general order of transformations is dilations, reflections and translations (DRT). However, with the use of matrices, as long as the matri calculations are completed in the order of the transformations, the correct equation or final result will e otained. Translations A translation is a transformation of a figure where each point in the plane is moved a given distance in a horizontal or vertical direction. It is when a figure is moved from one location to another without changing size, shape or orientation. Consider a marching and marching in perfect formation. As the leader of the marching and moves from a position P(, ) a steps across and steps up to a new position P (, ), all memers of the and will also move to a new position P ( + a, + ). Their new position could e defined as P (, ) = P ( + a, + ) where a represents the horizontal translation and represents the vertical translation. Topic 7 Matrices and applications to transformations 77

31 WOrKeD example 7 think If the leader of the marching and moves from position P(, ) to a new position P (, ), which is across and up steps, all memers of the marching and will also move the same distance and in the same direction. Their new position could e defined as: P'(', ') = P'( + a, + ) 4 P(, ) P ( +, + ) 5 a The matri transformation for a P(, ) translation can e given P P T = + a where a represents the horizontal translation and is the vertical translation. a The matri is called the translation matri and is denoted T. The translation matri maps the point P(, ) onto the point P (, ), giving the image point (, ) = ( + a, + ). A cclist in a iccle race needs to move from the front position at (0, 0) across positions to the left so that the other cclists can pass. Write the translation matri and find the cclist s new position. Write down the translation matri, T, using the information given. Appl the matri transformation for a translation equation. Sustitute the pre-image point into the matri equation. WritE 5 4 P(, ) P'(, ) The cclist moves across to the left units. Translating units to the left means each -coordinate decreases. T = 0 P P T = + 0 The pre-image point is (0,0). P P T 0 = Maths Quest MatheMatICaL MethODs VCe units and

32 4 State the cclist s new position calculating the coordinates of the image point from the matri equation. P P T 0 = = 0 The new position is (, 0). WOrKeD example 8 think translations of a shape Matri addition can e used to find the coordinates of a translated shape when a shape is moved or translated from one location to another on the coordinate plane without changing its size or orientation. Consider the triangle ABC with coordinates A(, ), B(0, ) and C(, ). It is to e moved units to the right and unit down. To find the coordinates of the vertices of the translated ΔA B C, we can use matri addition. First, the coordinates of the triangle ΔABC can e written as a coordinate matri. The coordinates of the vertices of a figure are arranged as columns in the matri. A B C 0 ΔABC = Secondl, translating the triangle units to the right means each -coordinate increases. Translating the triangle unit down means that each -coordinate decreases. The translation matri that will do this is. Finall, to find the coordinates of the vertices of the translated triangle ΔA B C add the translation matri 5 4 to the coordinate matri. A Aʹ A B C A B C B C Bʹ 0 + = Cʹ The coordinates of the vertices of the translated triangle ΔA B C = 4 are A (, ), B (, ) and C (, 0). 5 0 Find the translation matri if ΔABC with coordinates A(, ), B(0, ) and C(, ) is translated to ΔA B C with coordinates A (, 4), B (, ) and C (, ). Write the coordinates of ΔABC as a coordinate matri. WritE The coordinates of the vertices of a figure are arranged as columns in the matri. A B C ΔABC = 0 topic 7 MatrICes and applications to transformations 79

33 Write the coordinates of the vertices of the translated triangle ΔA B C as a coordinate matri. Calculate the translation matri using the matri equation: P = P + T 4 Translating the triangle units to the right means that each -coordinate increases. Translating the triangle unit up means that each -coordinate increases. Translations of a curve A translation of a curve maps ever original point (, ) of the curve onto a new unique and distinct image point (, ). Consider the paraola with the equation =. ΔA B C = A B C 4 4 = 0 + T T = 4 0 The translation matri is: T = If the paraola is translated units in the positive direction of the -ais (right), what is the image equation and what happens to the coordinates? As seen from the tale of values elow, each coordinate (, ) has a new coordinate pair or image = 0 point ( +, ). 9 8 (, ) = + = (, ) (, 9) + 9 (0, 9) 5 4 (, 4) + 4 (, 4) 4 (, ) + (, ) 0 0 (0, 0) (, 0) (, ) + (4, ) (, 4) + 4 (5, 4) 9 (, 9) + 9 (6, 9) The matri equation for the transformation for an point on the curve = can e written as: = + 0 The image equations for the two coordinates are = + and =. Rearranging the image equations to make the pre-image coordinates the suject, we get = + = and = (no change). To find the image equation, sustitute the image epressions into the pre-image equation. = = = = ( ) The image equation is = ( ). 80 Maths Quest MATHEMATICAL METHODS VCE Units and

34 WOrKeD example 9 Determine the image equation when the line with equation = + is transformed the translation matri T =. think WritE State the matri equation for the transformation given. = + State the image equations for the two coordinates. = + and = + Rearrange the equations to make the pre-image coordinates and the sujects. 4 Sustitute the image equations into the pre-image equation to find the image equation. 5 Graph the image and pre-image equation to verif the translation. WOrKeD example 0 think = and = = + = ( ) + = The image equation is =. 5 4 = + = Determine the image equation when the paraola with equation = is transformed the translation matri T =. State the matri equation for the transformation given. WritE = + State the image equations for the two coordinates. = and = + Rearrange the equations to make the pre-image coordinates and the sujects. 4 Sustitute the image epressions into the pre-image equation to find the image equation. = + and = = = ( + ) = ( + ) + The image equation is = ( + ) +. topic 7 MatrICes and applications to transformations 8

35 5 Graph the image and pre-image equation to verif the translation. Eercise 7.6 PRactise Work without CAS Consolidate Appl the most appropriate mathematical processes and tools Translations WE7 Find the image point for the pre-image point (, ) using the matri equation for translation = +. Find the image point for the pre-image point (, 4) using the matri equation for translation = +. WE8 Find the translation matri if ΔABC with coordinates A(0, 0), B(, ) and C(, 4) is translated to ΔA B C with coordinates A (, ), B (, ) and C (, ). 4 Find the translation matri if ΔABC with coordinates A(, 0), B(, 4) and C(, 5) is translated to ΔA B C with coordinates A (4, ), B (, 6) and C (, ). 5 WE9 Determine the image equation when the line with equation = is transformed the translation matri T =. 6 Determine the image equation when the line with equation = + is transformed the translation matri T =. 7 WE0 Determine the image equation when the paraola with equation = is transformed the translation matri T =. 8 Determine the image equation when the paraola with equation = + is transformed the translation matri T = 0. 9 A chess plaer moves his knight square to the right and squares up from position (, 5). Find the new position of the knight. 0 Find the image point for the pre-image point (, 0) using the matri equation for the translation = + 5. = ( + ) + 4 = Maths Quest MATHEMATICAL METHODS VCE Units and

36 MAstEr The image points are given = + and = +. Epress the transformation in matri equation form. 0 a On a Cartesian plane, draw ΔABC = 0 and ΔA B C = Find the translation matri if ΔABC = is translated to 0 0 ΔA B C = 0. Find the image equation when the line with equation = is transformed the translation matri T =. PAGE 7.7 Practice questionsonline Units & AOS Topic Concept 6 Reflections Concept summar Interactivit Refl ections int Find the image equation when the paraola with equation = is 7 transformed the translation matri T = 4. 5 Find the translation matri that maps the line with equation = onto the line with equation = +. 6 Find the translation matri that maps the paraola with equation = onto the paraola with equation = ( 7) +. 7 Write the translation equation that maps the paraola with equation = onto the paraola with equation = Write the translation equation that maps the circle with equation + = 9 onto the circle with equation ( ) + = 9. 9 Write the translation equation that maps the paraola with equation = onto the paraola with equation = ( a) +. 0 Write the translation equation that maps the circle with equation + = r onto the circle with equation ( a) + = r. Reflections A reflection is a transformation defined the line of reflection where the image point is a mirror image of the pre-image point. PROOFS Under a reflection, the image point P is a mirror image of the pre-image point P. The distances from the pre-image and image point to the reflection line are equal, with P and P on opposite sides of the reflection line. The line segment PP joining a point and its image is isected perpendicularl to the reflection line. The reflection line or reflecting surface is called the mediator. P(, ) Reflection line, M P'(', ') topic 7 MatrICes and applications to transformations 8

37 WOrKeD example think reflection in the -ais The reflection in the -ais maps the point P(, ) onto the point P (, ), giving the image point (, ) = (, ). The matri for reflection mapping in the -ais is: M = 0 0 In matri form, the reflection for an point in the -ais is: = 0 0 reflection in the -ais The reflection in the -ais maps the point P(, ) onto the point P (, ), giving the image point (, ) = (, ). The matri for reflection mapping in the -ais is: M = 0 0 In matri form, the reflection for an point in the -ais is: = 0 0 Find the image of the point (, ) after a reflection in: a the -ais the -ais. WritE a State the reflection matri to e used. a M = 0 0 Use the matri equation for reflection in the -ais. Sustitute the pre-image point into the matri equation. 4 Calculate the coordinates of the image point. 5 4 P'(', ') = (, ) P(, ) = (, ) P(, ) = ( 4, ) P'(', ') = (4, ) 4 5 = 0 0 The pre-image point is (, ). = 0 0 = 0 0 = The image point is (, ). State the reflection matri to e used. M = 0 0 Use the matri equation for reflection in the -ais. = Maths Quest MatheMatICaL MethODs VCe units and

38 Sustitute the pre-image point into the matri equation. 4 Calculate the coordinates of the image point. The pre-image point is (, ). = 0 0 = 0 0 = The image point is (, ). WOrKeD example think Find the image equation after = ( + ) is reflected in the -ais. WritE State the matri equation for reflection in the -ais. M = 0 0 = 0 0 Find the image coordinates. = = Rearrange the equations to make the pre-image = coordinates and the sujects. = 4 Sustitute the image epressions into the pre-image equation = ( + ) to find the image equation. 5 Graph the image and the pre-image equation to verif the reflection. 6 Alternativel: State the matri equation for reflection in the -ais. = ( + ) = ( + ) = ( ) The image equation is = ( ). = ( + ) = ( ) = Find the pre-image coordinates using the inverse of the transformational matri. X = T = 0 0 topic 7 MatrICes and applications to transformations 85

39 8 Multipl and simplif the matri equation. = = 9 Sustitute the image epressions into the pre-image equation = ( + ) to find the image equation. WOrKeD example reflection in the line with equation = The reflection in the line = maps the point P(, ) onto the point P (, ), giving the image point (, ) = (, ). The matri for reflection mapping in the line = is M = = 0 0. In matri form, the reflection for an point in the line = is = 0 0. = ( + ) = ( + ) = ( ) The image equation is = ( ). P'(', ') = (, 4) 5 4 = P(, ) = (4, ) Find the image of the point (, ) after a reflection in the line with equation =. think WritE State the reflection matri to e used. M = = 0 0 Use the matri equation for a reflection aout = 0 the line with equation =. 0 Sustitute the pre-image point into the matri equation. 4 Calculate the coordinates of the image point. reflection in a line parallel to either ais To determine the image point P (, ) from a reflection in a line parallel to either the -ais or the -ais, we need to consider the distance etween the point P(, ) and the parallel line. If we consider the distance from the -coordinate of P to the vertical reflection line as PD = a, to otain the image -coordinate we need to add the distance to the value of the mediator line, giving = a + a = a. The pre-image point is (, ). = 0 0 = 0 0 = The image point is (, ). 86 Maths Quest MatheMatICaL MethODs VCe units and

40 WOrKeD example 4 The reflection in the line = a maps the point P(, ) onto the point P (, ), giving the image point (, ) = (a, ). In matri form, the reflection for an point in the line = a is: = a 0 The reflection in the line = maps the point P(, ) onto the point P (, ) giving the image point (, ) = (, ). In matri form, the reflection for an point in the line = is: = A summar of the matrices for reflections are shown in the following tale. Reflection in Matri Matri equation -ais M = 0 = ais M = 0 0 line = M = = 0 0 line = a line = = 5 4 P(, ) = (, ) P(, ) = (, ) = 0 0 P'(', ') = (5, ) = 4 5 P'(', ') = (, ) = (, 5) = 0 0 = a 0 = Find the image of the point (, ) after a reflection in the line with equation: a = =. think a State the matri equation to e used. WritE a The point is reflected in the line =. = a 0 topic 7 MatrICes and applications to transformations 87

41 Sustitute the pre-image point and the value of a into the matri equation. Calculate the coordinates of the image point. State the matri equation to e used. Sustitute the pre-image point and the value of into the matri equation. Calculate the coordinates of the image point. Eercise 7.7 PRactise Work without CAS Consolidate Appl the most appropriate mathematical processes and tools Reflections WE Find the image of the point (, ) after a reflection in the -ais. Find the image of the point (5, ) after a reflection in the -ais. WE Find the image equation after = ( ) is reflected in the -ais. 4 Find the image equation after = + is reflected in the -ais. 5 WE Find the image of the point (, 5) after a reflection in the line with equation =. 6 Find the image of the point (8, ) after a reflection in the line with equation =. 7 WE4 Find the image of the point (, ) after a reflection in the line with equation =. 8 Find the image of the point ( 4, ) after a reflection in the line with equation =. 9 Find the image of the point (, 5) after a reflection in: a the -ais The pre-image point is (, ) and the value of a = from the line =. = = = The image point is (, ). The point is reflected in the line =. = The pre-image point is (, ) and the value of = from the line =. = = = The image point is (, ). the -ais. 0 Find the image of the point (8, 4) after a reflection in: a M M. Find the image of the point (9, 6) after a reflection in the line with equation =. Find the image of the point (0, ) after a reflection in the line with equation =. 88 Maths Quest MATHEMATICAL METHODS VCE Units and

42 Master 7.8 Units & AOS Topic Concept 7 Dilations Concept summar Practice questions Interactivit Dilations int-697 Find the image of the point (, ) after a reflection in the line with equation: a = =. 4 Find the image of the point (7, ) after a reflection in the line with equation: a = 4 =. 5 A point P is reflected in the line = to give an image point P (, 5). What are the coordinates of P? 6 A point P is reflected in the line = to give an image point P (, ). What are the coordinates of P? 7 The line with equation = + is transformed according to the matri equations given. Find the equation of the image for each transformation. a = 0 = c The line with equation = 4 8 The paraola with equation = + + is transformed according to the matri equations given. Find the equation of the image for each transformation. a = 0 = 0 c = Find the final image point when point P(, ) undergoes two reflections. It is firstl reflected in the -ais and then reflected in the line =. 0 Find the coordinates of the vertices of the image of pentagon ABCDE with A(, 5), B(4, 4), C(4, ), D(, 0) and E(0, ) after a reflection in the -ais. Dilations A dilation is a linear transformation that changes the size of a figure. The figure is enlarged or reduced parallel to either ais or oth. A dilation requires a centre point and a scale factor. A dilation is defined a scale factor denoted k. If k >, the figure is enlarged. If 0 < k <, the figure is reduced. Original Reduction Enlargement One-wa dilation A one-wa dilation is a dilation from or parallel to one of the aes. Dilations from the -ais or parallel to the -ais A dilation in one direction from the -ais or parallel to the -ais is represented the matri equation: = k 0 0 = k P(, ) P'(', ') = (k, ) where k is the dilation factor. 0 Topic 7 Matrices and applications to transformations 89

43 WOrKeD example 5 think The points (, ) are transformed onto points with the same -coordinate ut with the -coordinate k times the distance from the -ais that it was originall. The point moves awa from the -ais in the direction of the -ais a factor of k. This determines the horizontal enlargement of the figure if k > or the horizontal compression if 0 < k <. Dilations from the -ais or parallel to the -ais A dilation in one direction from the -ais or parallel to the -ais is represented the matri equation: = 0 0 k = k P'(', ') = (, k ) P(, ) where k is the dilation factor. 0 The point moves awa from the -ais in the direction of the -ais a factor of k. This determines the vertical enlargement of the figure if k > or if 0 < k <, the vertical compression. Find the coordinates of the image of the point (, ) under a dilation of factor from the -ais. WritE State the dilation matri to e used. 0 0 k Use the matri equation for dilation sustituting the value of k. Sustitute the pre-image point into the matri equation. 4 Calculate the coordinates of the image point. WOrKeD example 6 The dilation factor is k =. = 0 0 The pre-image point is (, ). = 0 0 = 0 0 = The image point is (, ). 6 Find the image equation when the paraola with equation = is dilated a factor of from the -ais. think State the matri equation for dilation. Find the equation of the image coordinates in terms of the pre-image coordinates. WritE = 0 0 = and = = 90 Maths Quest MatheMatICaL MethODs VCe units and

44 Rearrange the equations to make the pre-image coordinates and the sujects. 4 Sustitute the image values into the pre-image equation to find the image equation. 5 Graph the image and the pre-image equation to verif the translation. Two-wa dilations A dilation parallel to oth the -ais and -ais can e represented the matri equation: = k 0 0 k = k k where k and k are the dilation factors in the -ais and -ais directions respectivel. When k k the oject is skewed. When k = k = k the size of the oject is enlarged or reduced the same factor, and the P'(', ') = (k, k) matri equation is: P(, ) = k 0 0 k = k 0 0 = = = = ( ) 4 and = The image equation is = 4. 5 = = 4 = ki = k = k k where k is the dilation factor. Topic 7 Matrices and applications to transformations 9

45 WOrKeD example 7 think a Draw a diagram to represent this situation. State the coordinates of the vegetale patch as a coordinate matri. State the dilation matri. Jo has fenced a rectangular vegetale patch with fence posts at A(0, 0), B(, 0), C(, 4) and D(0, 4). a She wants to increase the size of the vegetale patch a dilation factor of in the -direction and a dilation factor of.5 in the -direction. Where should Jo relocate the fence posts? Jo has noticed that the vegetale patch in part (a) is too long and can onl increase the vegetale patch size a dilation factor of in oth the -direction and the -direction. Where should she relocate the fence posts? Will this give her more area to plant vegetales? Eplain. 4 Multipl the dilation matri the coordinate matri to calculate the new fence post coordinates. WritE a 5 4 D C A B 4 5 The coordinates of the vegetale patch ABCD can e written as a coordinate matri = The new fence posts are located at A (0, 0), B (9, 0), C (9, 6) and D (0, 6). State the coordinates of the vegetale patch as a coordinate matri. The coordinates of the vegetale patch ABCD can e written as a coordinate matri: State the dilation matri. The dilation matri is 0 0. Calculate the new fence post coordinates A B C D multipling the dilation matri the coordinate matri = = The new fence posts are located at A (0, 0), B (6, 0), C (6, 8) and D (0, 8). 9 Maths Quest MatheMatICaL MethODs VCe units and

46 4 Draw a diagram of the original vegetale patch, and the two transformed vegetale patches on the same Cartesian plane. 5 Determine the area for each vegetale patch. Eercise 7.8 PRactise Work without CAS Consolidate Appl the most appropriate mathematical processes and tools Dilations 0 9 Part () 8 Part (a) D 4 C A B The vegetale patch size when dilated a factor of in the -direction and a dilation factor of.5 in the -direction gives an area of 54 units. When dilated a factor of in oth the -direction and the -direction, the vegetale patch has an area of 48 units. The farmer will have less area to plant vegetales in the second option. WE5 Find the coordinates of the image of the point (, ) under a dilation of factor from the -ais. Find the coordinates of the image of the point (, 4) under a dilation of factor from the -ais. WE6 Find the image equation when the paraola with equation = is dilated a factor of from the -ais. 4 Find the image equation when the paraola with equation = is dilated a factor of from the -ais. 5 WE7 A farmer has fenced a vegetale patch with fence posts at A(0, 0), B(, 0), C(, 4) and D(0, 4). She wants to increase the vegetale patch size a dilation factor of.5 in the -direction and a dilation factor of in the -direction. Where should she relocate the fence posts? 6 Jack wants to plant flowers on a flower patch with corners at A(, ), B(4, ), C(, ) and D(, ). He wants to increase the flower patch size a dilation factor of in oth the -direction and the -direction. Where should he relocate the new corners of the flower patch? 7 Find the image of (, 5) after a dilation of parallel to the -ais. 8 Find the image of (, 4) after a dilation of parallel to the -ais. 9 A man standing in front of a carnival mirror looks like he has een dilated times wider. Write a matri equation for this situation. Topic 7 Matrices and applications to transformations 9

47 Master 7.9 Units & AOS Topic Concept 8 Cominations of transformations Concept summar Practice questions ONLINE 0 A transformation T is given = 0 0. a Find the image of the point A(, ). Descrie the transformation represented T. Find the image equation when the line with equation + = is 0 dilated 0. Find the image equation when the paraola with equation = is dilated 0 0. Find the image equation when the hperola with equation = is dilated + a factor of from the -ais. 4 The equation = is transformed according to = 0 0. a What is the mapping produced in the matri transformation? What is the image equation? 5 Find the image equation when the circle with equation + = 4 is transformed according to = The coordinates of ΔABC can e written as a coordinate matri 0. It has undergone a transformation T given = k 0 0. a Find the dilation factor, k, if the image coordinate point A is (, 0). Calculate the coordinates of the vertices of ΔA B C. 7 Find the factor of dilation when the graph of = is otained dilating the graph of = from the -ais. 8 a Find the image equation of + = under the transformation dilation a factor of parallel to the -ais. Is there an invariant point? Cominations of transformations A comined transformation is made up of two or more transformations. Doule transformation matrices If a linear transformation T of a plane is followed a second linear transformation T, then the results ma e represented a single transformation matri T. PAGE PROOFS When transformation T is applied to the point P(, ) it results in P (, ). When transformation T is then applied to P (, ) it results in P (, ). Summarising in matri form: = T = T 94 Maths Quest MATHEMATICAL METHODS VCE Units and

48 WOrKeD example 8 think Sustituting T for into = T results in = T T. To form the single transformation matri T, the first transformation matri T must e pre-multiplied the second transformation matri T. This is written as: T = T T Common transformation matrices used for cominations of transformations M = 0 0 M = 0 0 M = = 0 0 D k, = k 0 0 Reflection in the -ais Reflection in the -ais Reflection in the line = Dilation in one direction parallel to the -ais or from the -ais D, k = 0 0 k Dilation in one direction parallel to the -ais or from the -ais D k, k = k 0 Dilation parallel to oth the - and -ais 0 k (k and k are the dilation factors) Determine the single transformation matri T that descries a reflection in the -ais followed a dilation of factor from the -ais. Determine the transformation matrices eing used. State the comination of transformations matri and simplif. WritE T = reflection in the -ais T : M = 0 0 T = dilation of factor from the -ais T : D, = 0 0 T = T T T = D, M T = 0 0 = State the single transformation matri. The single transformation matri is: T = 0 0 topic 7 MatrICes and applications to transformations 95

49 EErcisE 7.9 PrActisE Work without cas consolidate Appl the most appropriate mathematical processes and tools MAstEr Cominations of transformations WE8 Determine the single transformation matri T that descries a reflection in the -ais followed a dilation factor of from the -ais. Determine the single transformation matri T that descries a reflection in the line = followed a dilation of factor from oth the and -ais. A rectangle ABCD is transformed under the transformation matri T = 5 8, to give vertices at A (0, 0), B (, 0), C (, ) and D (0, ). a Find the vertices of the square ABCD. Calculate the area of the transformed figure ABCD. 4 a Find the image point of point P (, ) when the point P(, ) undergoes a doule transformation: a reflection in the -ais followed a translation of 4 units in the positive direction of the -ais. Reverse the order of the pair of transformations in part a. Is the image different? 5 State the image of (, ) for a translation of followed a reflection in the -ais. 6 Descrie full a sequence of two geometrical transformations represented T = The triangle ABC is mapped the transformation represented T = onto the triangle A B C. Given that the area of ABC is 0 units, find the area of A B C. 8 a State the transformations that have undergone T = Determine the image of the curve with equation =. 9 a State the transformations that have undergone T = 0 0 Determine the image of the curve with equation = Find the image equation of = under a doule transformation: a reflection in the -ais followed a dilation factor of parallel to oth the - and -ais. Find the image equation of = under a doule transformation: a reflection in the -ais followed a dilation of parallel to the -ais. A rectangle ABCD with vertices at A(0, 0), B(, 0), C(, ) and D(0, ) is transformed under the transformation matri T =. Find the new area of the transformed rectangle. If D k denotes a dilation factor of k parallel to oth aes, what single dilation would e equivalent to D k? 4 Check whether the transformation a reflection in the -ais followed a reflection in the line = is the same as a reflection in the line = followed a reflection in the -ais. 96 Maths Quest MatheMatICaL MethODs VCe units and

50 ONLINE ONLY 7.0 Review the Maths Quest review is availale in a customisale format for ou to demonstrate our knowledge of this topic. the review contains: short-answer questions providing ou with the opportunit to demonstrate the skills ou have developed to efficientl answer questions without the use of CAS technolog Multiple-choice questions providing ou with the opportunit to practise answering questions using CAS technolog ONLINE ONLY Activities to access ebookplus activities, log on to Interactivities A comprehensive set of relevant interactivities to ring difficult mathematical concepts to life can e found in the Resources section of our ebookplus. Etended-response questions providing ou with the opportunit to practise eam-stle questions. A summar of the ke points covered in this topic is also availale as a digital document. REVIEW QUESTIONS Download the Review questions document from the links found in the Resources section of our ebookplus. studon is an interactive and highl visual online tool that helps ou to clearl identif strengths and weaknesses prior to our eams. You can then confidentl target areas of greatest need, enaling ou to achieve our est results. Units & Matrices and applications to transformations Sit topic test topic 7 MatrICes and applications to transformations 97