1.   (a) A furniture store sells chairs, small tables and large tables.  Each chair costs 。10, each small table costs 。20, and each large table costs 。40.

To prepare for a big event, a customer goes to the store and buys  100 items, including 70 tables. They spend a total of 。2500.

Write this problem as a system of simultaneous linear equations and write down the corresponding augmented matrix. A solution is not required. [2 marks]

(b) Write a new version of the question in part (a) with the numbers of items changed (but the same prices), ensuring that the solutions to the equations are still positive whole numbers. Justify your answer. [3 marks]

2. A 3 × 3 square matrix A is said to be upper triangular if A21   = 0, A31   = 0 and A32  = 0.

(a)  Give an example of a 3 × 3 matrix which is upper triangular and in echelon form, but not in reduced echelon form. Justify your answer. [3 marks]

(b)  Explain how an upper triangular 3 × 3 matrix with non-zero entries on its leading diagonal (the diagonal from top left to bottom right) can be brought to reduced echelon form. [4 marks]

3.   (a) Suppose that a and b are positive real numbers and b > 2a.  Sketch the feasible set given by the inequalities:

y < x + a,    x + 2y < b,    x > 0,    y > 0,

and find its vertices. Find the maximum value of x + y in the feasible set in terms of a and b. [4 marks]

(b)  Find the maximum value of x + 2y in the feasible set in the previous question, and find all of the points in the feasible set where this value is obtained. [4 marks]

4.   (a)  Find all of the 2 × 2 matrices X satisfying X2  = X for which all of the entries of X are equal. [4 marks]

(b)  Let X =  ╱c(a)   d(b)、  and Y =  ╱g(e)   h(f)、  be 2 × 2 matrices with real entries.  Show

that the determinant of XY is equal to the product of the determinants of X and Y . [4 marks]

(c)  Give an example of a 3 × 3 matrix whose only eigenvalues are 1 and 2. [4 marks]

5.   (a)  Find a vector v in R3  such that {(1, 1, 0), (1, 2, 0), v} is a basis of R3 , justifying your answer. [4 marks]

(b)  Give an example of three vectors v1 , v2 , v3  which are linearly dependent, but for which v1  and v2  are linearly independent. [4 marks]

6.   (a) The trace of a 2 × 2 matrix A =  、  with real entries is given by the formula:

Tr(A) = a11 + a22 .

Let X =  ╱c(a)   d(b)、  and Y =  ╱g(e)   h(f)、  be 2 × 2 matrices with real entries.  Verify

that

Tr(XY) = Tr(YX).    [4 marks]

(b)  Let M2 (R) denote the set of all 2 × 2 matrices with real entries.  If λ e R and

A = ╱c(a)   d(b)、  e M2 (R), let λA denote the matrix  、.

A subset V of M2 (R) is said to be a subspace of M2 (R) if the zero matrix lies in V , the sum of any two matrices in V lies in V , and λA lies in V for any matrix A in V and λ e R.

Show that the set

T = {A e M2 (R) : Tr(A) = 0}

is a subspace of M2 (R). [4 marks]

7.   (a)  Find two invertible 2 × 2 matrices A and B with positive integer entries such that A + B is not invertible. [4 marks]

(b) A certain city has two key industries:  electricity and steel.  To produce 。1 worth of electricity takes 。a worth of electricity and 。c worth of steel, and to produce

。1 worth of steel takes 。b worth of electricity and 。d worth of steel. We assume that 0 < a < 1, 0 < b < 1, 0 < c < 1, and 0 < d < 1.

Use the Leontief model to work out how much each industry should produce to allow for consumption of 。p worth of electricity and 。q worth of steel, in terms of a, b, c, d, p and q, where p and q are positive real numbers.

Your answer should  include clear statements of the  input-output  matrix  (using the ordering Electricity, Steel) and the demand matrix.  It should include a clear explanation of what you are doing and have a clear conclusion.  You should also include any additional assumptions on a, b, c and d that you need. [8 marks]

8.   (a)  Let a be a real number with 0 < a < 3.  Solve the following linear programming

problem using the simplex method (as given in the course):

Maximise 2x + y subject to the constraints:

ax + ay < 6;

-x + y < 2;

x - y < 2;

x > 0, y > 0.

Your answer should be given in terms of a, and should include the following:        ● A clear and full statement of an equivalent problem with slack variables and additional variable (objective function) introduced;

● The initial simplex tableau;

● A clear justification of each step in the algorithm, including an explanation of your choice of pivot;

● A clear conclusion, including a list of the Type I and Type II variables and how they are used to find the solution; [6 marks]

(b) Suppose the constraint -x + y < 2 is replaced by -x + y < 2 + h where h is a real number.  Determine for which values of h a marginal analysis approach (without any further pivoting calculations) is valid. For the valid values of h, use a marginal analysis approach to compute the new solution to the problem. [4 marks]

(c)  Repeat part (a) with the assumption that a > 3 instead of 0 < a < 3. [6 marks]

(d) Suppose instead that a < 0 in part (a). Why does the simplex method fail in this case? Explain this by sketching the feasible set.  [4 marks]

9.   (a)  Give an example of a regular stochastic matrix which is not absorbing. Justify your answer. [4 marks]

(b)  Give an example of an absorbing stochastic matrix which is not regular.  Justify your answer. [4 marks]