Barry University MAT-108 Take-home Test 5 (Sections 6.1, 6.2, 6.3, 7.1 & 7.2) Student Name: 1) Construct the matrix 𝐴𝐴 = �𝐴𝐴𝑖𝑖𝑖𝑖 � if 𝐴𝐴 is 3𝑥𝑥2 and 𝐴𝐴𝑖𝑖𝑖𝑖 = −2𝑖𝑖 + 𝑗𝑗. 2) If 𝐴𝐴 is a 7𝑥𝑥4 matrix , what is the size of 𝐴𝐴𝑇𝑇 ?. 2 3 1 1 2 2 0 1 3) 𝐿𝐿𝐿𝐿𝐿𝐿 𝐴𝐴 = � 1 0 2 � , 𝐵𝐵 = � � , 𝐶𝐶 = � 3 0 � then find: −1 1 0 −2 1 −3 −1 −2 i. ii. iii. iv. v. 𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵 𝐶𝐶𝐶𝐶 𝐵𝐵𝐵𝐵 𝐶𝐶𝐶𝐶 2 −1 4) If 𝐴𝐴 = � � and 𝐵𝐵 𝑖𝑖𝑖𝑖 𝑡𝑡ℎ𝑒𝑒 2𝑥𝑥2 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡ℎ𝑒𝑒𝑒𝑒 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓: 3 −2 i. ii. iii. iv. v. 𝐴𝐴 + 2𝐵𝐵 −3𝐴𝐴 – 𝐵𝐵 𝐴𝐴𝑇𝑇 (𝐴𝐴 − 𝐵𝐵)𝑇𝑇 𝑥𝑥11 5) 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑥𝑥11 , 𝑥𝑥12 𝑎𝑎𝑎𝑎𝑎𝑎 𝑥𝑥22 𝑖𝑖𝑖𝑖 𝑋𝑋 = � −8 𝑥𝑥12 𝑥𝑥22 �, 2 −3 4 1 𝐴𝐴 = � � 𝑎𝑎𝑎𝑎𝑎𝑎 𝐵𝐵 = � � 𝑎𝑎𝑎𝑎𝑎𝑎 𝐴𝐴𝐴𝐴 = 𝐵𝐵 −1 1 2 −2 6) Suppose the price (𝑖𝑖𝑖𝑖 $) matrix for products 𝑋𝑋, 𝑌𝑌 𝑎𝑎𝑎𝑎𝑎𝑎 𝑍𝑍 is giving by 𝑃𝑃 = (100 250 150). If the quantities (in units) of 𝑋𝑋, 𝑌𝑌 𝑎𝑎𝑛𝑛𝑛𝑛 𝑍𝑍 that are 700 purchased are giving by the column matrix 𝑄𝑄 = �200�, what is the total 400 cost of the purchase? 2 −2 0 1 7) 𝐿𝐿𝐿𝐿𝐿𝐿 𝐴𝐴 = � � 𝑎𝑎𝑎𝑎𝑎𝑎 𝐵𝐵 = � � 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓: −1 3 1 3 i. ii. iii. (𝐴𝐴𝐴𝐴)𝑇𝑇 𝐵𝐵𝑇𝑇 𝐴𝐴𝑇𝑇 𝑊𝑊ℎ𝑎𝑎𝑎𝑎 𝑖𝑖𝑖𝑖 𝑡𝑡ℎ𝑒𝑒 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑠𝑠ℎ𝑖𝑖𝑖𝑖 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 (𝐴𝐴𝐴𝐴)𝑇𝑇 and 𝐵𝐵𝑇𝑇 𝐴𝐴𝑇𝑇 1 0 8) 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝐴𝐴3 if 𝐴𝐴 = � � 1 2 9) 𝐿𝐿𝐿𝐿𝐿𝐿 𝑋𝑋 = (𝑥𝑥 𝑦𝑦 𝑧𝑧)𝑇𝑇 . Write the following system of equations using matrix multiplication 𝐴𝐴𝐴𝐴 = 𝐵𝐵 𝑎𝑎𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑖𝑖𝑖𝑖. 10) 11) 3𝑥𝑥 2 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑋𝑋 𝑖𝑖𝑖𝑖 𝑑𝑑𝑑𝑑𝑑𝑑 � 𝑥𝑥 − 2 2𝑥𝑥 − 𝑦𝑦 + 3𝑧𝑧 = 3 �5𝑥𝑥 − 𝑦𝑦 + 2𝑧𝑧 = 20 3𝑥𝑥 − 2𝑦𝑦 + 2𝑧𝑧 = 9 −4 1 3 �=3 Maximize 𝑃𝑃(𝑥𝑥, 𝑦𝑦) = 3𝑥𝑥 + 2𝑦𝑦 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 12) 𝑥𝑥 + 𝑦𝑦 ≤ 70 𝑥𝑥 + 3𝑦𝑦 ≤ 240 � 𝑥𝑥 + 3𝑦𝑦 ≤ 90 𝑥𝑥, 𝑦𝑦 ≥ 0 CANOIL, a canola oil manufacturer, needs nitrogen (𝑁𝑁), phosphate (𝑃𝑃) and potassium (𝐾𝐾) to make its product. The minimum requirements for a major order are 1000 kg of 𝑁𝑁, 1600 kg of 𝑃𝑃 and 1500 kg of 𝐾𝐾. Company 𝐴𝐴 supplies containers costing $210 each of which consists of 5 kg of 𝑁𝑁, 4 kg of 𝑃𝑃 and 3 kg of 𝐾𝐾. How does CANOIL minimize cost? 5 Company 𝐵𝐵 Min. kg reqd Nitrogen Company 𝐴𝐴 2 1000 Phosphate 4 4 166 Potassium 3 5 1500 $ 120 $ 90 Hint: minimize cost function 𝑐𝑐(𝑥𝑥, 𝑦𝑦) = 120𝑥𝑥 + 90𝑦𝑦.
Purchase answer to see full attachment