QUESTION 5

(a) Draw a digraph with three or four vertices and at least four edges. Do not choose an example from lectures or workshops. (i) Construct the adjacency matrix M for the digraph that you have drawn.

(ii) Calculate the reachability matrix M*, explaining all the steps in your solution. (iii) Interpret your reachability matrix M* in terms of your digraph. I.e., what does the reachability matrix M* tell you about your digraph?

(b) Write down a 2x2 matrix S whose entries are integers, all different, between -3 and 7 inclusive. One of the entries must be 5.

(i) A square has vertices A(1,-1), B(1,1), C(-1,1) and D(-1,-1). Find the images A, B, C and D of each of these vertices under the transformation S.

(ii) Plot the transformed square and describe the geometric transformation which has taken place.

(c) (i) Construct a 2x2 matrix T whose entries are non-zero integers and whose determinant is 15. Explain how you constructed your matrix.

(ii) Find the inverse of your matrix T.

(iii) Find the product ST where S is your matrix from (b).

(iv) Write down the system of equations represented by the matrix equation ( ) = ( 3 1 ).

Use your answer to (ii) to find the solution to the system.