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TUTORIAL MATRICES AND SYSTEMS OF LINEAR EQUATIONS 1. 5 1 1 1 1 3 0 Given that A = , B = ,C= 4 3 and D = 0 2 3 6 2 5 2 3 4 0 2 4 3 1 3 2 evaluate the following; 2. 3. (a) BC + 2A (b) D2 – CB (c) A – 3(BC + A) 5 0 0 Given two matrices A = 1 8 0 , B = 1 3 5 2 0 0 1 5 0 1 3 2 (a) Determine if AB = BA (b) Find m and n such that A = mB + nI3. 2 3 The matrices A and B are given as A = , B= 1 1 a b 1 1 where a and b are real numbers. Find the diagonal matrix D such that ADA-1 = B. 4. 1 1 1 Given that matrix P = 1 0 1 , find the matrix A such that 0 1 2 2 0 0 P AP = 0 2 0 . 0 0 4 -1 5. Find the inverse matrix of the following using; (i) The adjoint method (ii) The Gauss-Jordan Elimination method (a) 6. 0 2 1 4 1 5 1 1 2 (b) 1 1 1 Given that matrix A = 1 0 2 and B = 2 3 7 2 4 2 4 5 3 1 1 1 4 2 6 3 5 1 . Prove that AB = kI, 3 1 1 where I is an identity matrix and k is a constant. 7. 2 1 2 Show that A = is a zero of polynomial f(A) = A – 5A + I. 5 3 Hence find the matrix A-1 and A3 8. 1 2 1 Given that matrix A = 1 1 0 , find the matrix A2. If A3 – A2 – 6A = I, 3 1 1 where I is an identity matrix , find the inverse matrix of A. 9. 2 1 1 If P = 1 0 1 and Q = 3 1 4 3 1 1 1 5 1 , find PQ and 1 1 1 deduce the matrix P-1. Express the following system in its matrix form. 2x y z 3 x z 1 3x y 4 z 0 Hence solve the linear system. 10. Solve the following systems of equation system by using, (a) The Inverse matrix : x 2y z 5 2 x 3 y 2 z 1 x 3y 2z 2 (b) Cramer’s rule : x 3y 2z 1 5 x 2 y 7 z 2 x 2y z 2 (c) The Gauss-Jordan Elimination : 2 x 3 y 4 z 1 2 x 2 y 3z 2 x 2 y 2 z 3 11. 1 4 2 3 2 2 2 , find MN. Hence find M-1. If M = 4 1 2 and N = 8 5 2 1 5 2 1 3 In a supermarket, there were three promotion packages A, B and C which offer shirts, long trousers and neckties. The number of each item and the promotion price for each package is given in the following table: Package No. of No. of No. of Promotional price Shirts trousers neckties (RM) A 3 2 2 256 B 4 1 2 218 C 2 1 3 173 By using x, y and z respectively to represent the price of a shirt, a pair of trousers and a necktie, write down a matrix equation to represent the information above. Solve the equation to determine the price of each item. 12. Three weight lifters, Herman, Thor and Akiro entered a weight-lifting contest in which they could make up their total weight by adding three different sized weights. Herman used 4 of weight X, 3 of weight Y and 5 of weight Z for a total lift of 295kg. Thor used 2 of weight X, 5 of weight Y and 3 of weight Z for a total lift of 295kg. Akiro used 3 of weight X, 4 of weight Y and 5 of weight Z for a total lift of 310kg (a) Use the above information to determine the three separate weights by using elementary row operation method. (b) If Thor replaced one weight Z by a weight Y, would this change the result of the competition? 13. Use the quadratic function y = ax2 + bx + c to model the following data : x (Age of a driver) y (Average number of automobile accidents per day in the United States) 20 400 40 150 60 400 Use Cramer’s Rule to determine values for a, b, and c. 14. A mixed nut company uses Cashews nuts, Macadamias nuts and Brazil nuts to make 3 gourmet mixes. The table below indicates the weight in hundreds of gram of each kind of nut required to make a kilogram of mix. Cashews nuts Macadamias nuts Brazil Nuts Mix A 5 3 2 Mix B 2 4 4 Mix C 6 1 3 If 1 kg of mix A costs $12.50 to produce, 1 kg of mix, B costs $12.40 and 1 kg of mix C costs $11.70, determine the cost per kilogram of each different kind of nut. Hence, find the cost per kg to produce a mix containing 400 gram of Cashews nuts, 200 gram of Macadamias nuts and 400 grams of Brazil nuts. ANSWERS 1. (a) 9 2 11 3 2. (a) AB = BA 3. 3 0 0 2 4. 1 3 3 A = 3 5 3 6 6 4 5. (a) 7 1 13 27 3 4 3 6. 7. 3 1 A-1 = 5 2 8. 6 1 2 A 0 3 1 , 7 6 4 2 (b) (b) m = -1 5 10 6 2 1 0 0 6 0 1 0 , k 6 0 0 1 31 1 2 22 27 12 10 19 17 43 24 A3 = 120 67 1 1 1 A 1 2 1 4 5 3 1 31 10 33 1 (c) n=3 (b) 8 1 1 30 9 2 4 6 22 14 6 9. 10. 1 0 0 PQ 20 1 0 , 0 0 1 P 1 1 3 1 1 1 5 1 , 2 1 1 1 (a) z 2 (c) 1 3 x2 x3 y 1 y 5 23 21 52 z 21 z 2 z 2 y 1 0 0 MN = 9 0 1 0 0 0 1 M 1 1 4 2 1 8 5 2 9 2 1 5 X = RM 30 12. y 1 (b) x 11. x3 Y = RM 68 (a) X = 25kg, Y = 40kg, Z = 15kg (b) Yes, Thor lifts 320kg 5 , b 50 , c 1150 8 13. a 14. Cashews $12 per kg Macadamias $15 per kg Brazil nuts $10 per kg $11.80 Z = RM 15