Matrices Practice

1 Secondary 4 Mathematics: Matrices 1. Definition of a Matrix • A matrix is a rectangular array of numbers arranged in rows and columns • Example ! 1 3 5 7 9 11 ' is a matrix with 3 rows and 2 columns. - We can say the matrix above has an order of 3 by 2 and is known as a 3 × 2 matrix. • In general, a m × n matrix has m rows and n columns. The numbers inside a matrix are called the elements of the matrix. 2. Addition and Subtraction of Matrices )𝑎 𝑏 𝑐 𝑑. ± 0𝑒 𝑓 𝑔 ℎ5 = 0𝑎 ± 𝑒 𝑏 ± 𝑓 𝑐 ± 𝑔 𝑑 ± ℎ5 Example ! 1 3 5 7 9 11 ' + ! 11 3 5 6 4 12 ' = ! 1 + 11 3 + 3 5 + 5 7 + 6 9 + 4 11 + 12 ' = ! 12 6 10 13 13 23 ' 3. Equal Matrices • Two matrices are equal if they have the same order and have the same corresponding elements. If A = )𝑎 𝑏 𝑐 𝑑., B = 0𝑒 𝑓 𝑔 ℎ5 and A = B, then )𝑎 𝑏 𝑐 𝑑. = 0𝑒 𝑓 𝑔 ℎ5 ∴ 𝑎 = 𝑒, 𝑏 = 𝑓, 𝑐 = 𝑔, 𝑑 = ℎ

2 4. Multiplication of a Matrix by a Real Number • When a matrix is multiplied by a real number k, the product will be the matrix with each of its elements multiplied by k. 𝑘 )𝑎 𝑏 𝑐 𝑑. = )𝑘𝑎 𝑘𝑏 𝑘𝑐 𝑘𝑑. Example 3 )2 4 6 1 5 9. = )2 × 3 4 × 3 6 × 3 1 × 3 5 × 3 9 × 3. = )6 12 18 3 15 27. 5. Multiplication of Two Matrices • For matrix multiplication of two two matrices A and B, the product AB is only possible when the number of columns of A = number of rows of B • The multiplication of its matrices is not commutative, i.e. AB ≠ BA • The multiplication of matrices is associative, i.e. A(BC) = AB(C)

3 Matrices (Worksheet 1) 1. Given that 𝑷 = )1 2 3 4. , 𝑸 = )2 2 6 4. and 𝑹 = )2 4 6 8., find (a) 𝑷 + 𝑹 Answer: (a) ________________ (b) 𝑹 − 𝑸 Answer: (b) ________________ (c) 2𝑹 − 𝑷 Answer: (c) ________________

4 2. Given that 𝑷 = )10 5 3 −2. , 𝑸 = )𝑥 1 0 −1. and 𝑹 = )15 𝑦 3 −1., find the values of x and y such that 𝑷 − 𝑸 = 𝑹. Answer: x = ________________ y = ________________ 3. Given that 𝑷 = )1 𝑥 5 0. , 𝑸 = 00 5 𝑦 15 and 𝑹 = )2 2 6 4., find the values of x and y and the scalar k such that 𝑷 + 2𝑸 = 𝑘𝑹. Answer: x = ________________ y = ________________ k = ________________

5 4. Given that 𝑨 = )−3 −1 1 0 . , 𝑩 = )−1 2 . and 𝑪 = (5 −1), find (a) AB, (b) BC, (c) 𝑨! Answer: (a) ________________ (b) ________________ (c) ________________

6 5. It is given that )3 −2 0 1 0 2. M 𝑎 9 𝑏 N = ) 6 −7.. Find the value of a and of b. Answer: a = ________________ b = ________________ 6. If h and k are constants such that ℎ )2 8. + 𝑘 )3 4. = )16 48., find the values of h and k. Answer: h = ________________ k = ________________

7 7. It is given that 𝑨 = (1 0 1) , 𝑩 = ! 1 −28 2 ' , 𝑪 = )1 −3 5 0 . and AB = 2D. Find (a) D (b) C2 Answer: (a) D = ________________ (b) C2 = ________________

8 8. Given that )2 𝑧 5 𝑥. = )𝑦 4 5 −2. )1 3 0 2. , find the values of x, y and z. Answer: x = ________________ y = ________________ z = ________________

9 9. A drink stall sells ice-lemon tea and barley drink, each available in small, regular and large cups. The number of cups of drinks sold over a 10-minute period is given in the table below. Size of cup Small Regular Large Ice-lemon tea 4 8 2 Barley 1 5 0 Price per cup $1.30 $1.50 $1.80 It is given that M = )4 8 2 1 5 0. and 𝑵 = ! 1.3 1.5 1.8 '. (a) (i) Find MN. (ii) Explain what your answer (a)(i) represents. Answer: (a)(i) MN = ________________ (a)(ii) ___________________________________________________________________ (b) (i) Given that 𝑷 = (1 1), find PM. (ii) Explain what your answer (b)(i) represents. Answer: (b)(i) PM = ________________ (b)(ii) ___________________________________________________________________

10 10. A toy factory makes toy cars and toy trains. The following table is used in calculating the cost of making each toy. Labour (hours) Metal Cast (pieces) Paint (tins) Car 2 3 2 Trains 3 P 1 Labour costs $8 per hour, metal cast costs $1 per piece and paint costs $q per tin. The information can be summarized in the matrices A and B, where A = 02 3 2 3 𝑝 15 and 𝑩 = ! 8 1 𝑞 '. (a) Given that AB = )29 35., find the value of p and of q. Answer: (a) p = ________________ q = ________________ (b) Evaluate (200 150) )29 35.. Answer: (b) ________________ (c) Explain what your answer to (b) represents. Answer : (c) ______________________________________________________________ ______________________________________________________________