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# Question 1 (a) Let A =µ3 1 4 2¶B =µ−1 1 2 −2¶and C =  1 2 −1 3 −2 1 

Question 1 (a) Let A =µ3 1 4 2¶B =µ−1 1 2 −2¶and C = 
1 2 −1 3 −2 1 
. Either evaluate, or explain why it is not possible to evaluate, each of the following matrices. (i) 2A−3B; (ii) B + C; (iii) AB; (iv) CTB; (v) ACT; (vi) BAC; (vii) A−1; (viii) B−1. [4]
(b) A market gardener has 500m2 of land available for planting asparagus, beetroot and cabbage. It costs \$9 per m2 to plant asparagus, \$4 per m2 to plant beetroot and \$6 per m2 to plant cabbage. The market gardener spends on average \$6 per m2 on the production of crops, and all of the land is used in the production. Find the possible areas of asparagus, beetroot and cabbage the market gardener may produce, stating clearly any restrictions on your solution. [3] (c) Let D =  1 −2 3 0 2 4 0 0 6 and E =  1 2 1 3 2 −7 2 4 2 . Write down thedeterminants of D and E. In each case, brieﬂy justify your answer. [1] (d) Let u =  2 1 3 and v =  2 7 −3  . (i) Find the cosine of the angle between u and v. [1] (ii) Find a basis for the orth

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