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Hand your completed assignment in to the correct box in the Student Resource Centre (G402 Building 301) before the due date. Individualised coversheets should be downloaded from Canvas and attached to your assignment. Instructions on how to do this can be found on Canvas. Remember that these are unique to you. You should not print o a coversheet for your friend as well. Show all working and note that late assignments will not be marked. This assignment was written by Jared Hockly (j.hockly@auckland.ac.nz). The tutors in the Math- ematics Assistance Room can give you help and advice, but all work submitted must be your own (Assistance Room Hours and Location: Tuesday - Friday, 10-4, in 302.170). Nga manaakitanga - Good luck. 1. The following vector equation usually describes a plane in R3 . x = u + t1v + t2w What conditions on the vectors u; v;w 2 R3, would create an object that is not a plane? Some types of vectors, and relationships between vectors that may be helpful to think about are: zero vectors, unit vectors, parallel vectors, orthogonal vectors and linear combinations. [3 marks] 2. The following augmented matrix represent a system of equations, where b and c are some real number. 2 4 0 4 b c 2 1 1 0 5 2 2 10 3 5 (a) Solve (in terms of b and c) the system of equations by transforming the matrix into reduced row echelon form. [3 marks] (b) For what values of b and c will the system of equations have: (i) A unique solution. (ii) An in nite number of solutions. (iii) No solutions. [4 marks] 3. Let A, B and C be the following matrices: A =  6 2 15 5  B = 0 @ 3 4 6 7 9 10 1 AC =  8 0 5 2 6 0  For each of the expressions below, decide whether or not it can be calculated. If it cannot be calculated explain why. If it can be calculated, give the result (with working). (a) AB (b) BT 􀀀 CT (c) BTB (d) A􀀀1 (e) B􀀀1 (f) BA [10 marks] 4. Consider the two lines in R3: L1 : (x; y; z) = (1; 2; 3) + t(3; 1; 2) L2 : (x; y; z) = (1; 0; 8) + s(1;􀀀2; 2) (a) Deduce whether the two lines (L1 and L2) are: parallel, orthogonal or neither. [2 marks] (b) Give the parametric equations for each of the lines. [1 marks] (c) Use the equations you formed in (b), to investigate whether these lines intersect at a point, or not. It is expected that you will work with an augmented matrix. [3 marks] (d) Show how you could change the position vector of L1 (i.e use a vector not equal to (1,2,3)) to get the opposite result to what you found in (c). Give two di erent examples of vectors, that would achieve this. [2 marks] (e) Attempt to give an equation of all position vectors (x0, y0, z0) that could be used in the equation of L1 in place of (1,2,3) to achieve the opposite result to (b). [3 marks] 5. Let us de ne a \Poutama1 matrix" , Pn as follows:  a square matrix (of size n by n) that has only 0's or 1's as its entries,  has a leading 1 at the far right of the top row,  each row lower down, has it's leading 1 placed one position further to the left,  directly below any leading 1, there is also a 1,  all other entries are 0. Here is P5 : 0 BBBB@ 0 0 0 0 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 1 CCCCA (a) Show what the matrix P3 would be, and calculate (P3)2. [2 marks] (b) Find the inverse of P5. [3 marks] (c) Find the matrix B, such that: P3B = 0 @ 1 3 5 3 5 5 5 5 3 1 A [3 marks] (d) Investigate the value(s) of det(Pn), justify any claims you make. [4 marks] 1\Poutama" is a Te Reo Maori word which can mean staircase and is a common pattern in Tukutuku panels 6. A business has asked for a solution to the following problem. They have 3 companies o ering cleaning services to them. All three charge the same cost, on the budget they have, they are able to purchase a total of 70 hours of cleaning from them. Let x1 be the number of hours they employ company one, x2 be the number of hours they employ company 2, and x3 be the number of hours they employ company 3. Then, x1 + x2 + x3 = 70 They have found the companies clean at di erent rates per hour. Company 1 cleans windows at 7m2 per hour and oors at 120m2 per hour, Company 2 cleans windows at 9m2 per hour and oors at 100m2 per hour, company 3 cleans windows at 9:5m2 per hour and oors at 90m2 per hour. The Business currently needs 545m2 of windows cleaned and 7800m2 of oors cleaned, but these values could change a small amount in the future. (a) Write the other 2 equations for this situation. [1 marks] (b) Solve the system of equations by using an inverse matrix. [4 marks] (c) Give the bene t(s) of using an inverse matrix, over row operations on an augmented matrix, for this situation. [2 marks]

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