As usual start Maple and load all the linear algebra progams.

The matrix underlying the recurrence c(n+1)= 2*c(n) + 115*c(n-1) + 364*c(n-2) is
> C:=<<2,1,0>|<115,0,1>|<364,0,0>>;

Diagonalise C utilising
> (v,P):=Eigenvectors(C);
and notice that its eigenvectors have the special "powers
of eigenvalues" formula they are supposed to have.

Form Ck as the kth power of C by diagonalisation and now suppose that the 
first three values in the sequence are c(0)=10, c(1)=7 and c(2)=-17. 
Multiply Ck by the initial values vector, remembering to use the correct order.

Calculate c(3) and c(4) using the recurrence then use the power matrix to
get c(k) and check with your values with k=3 and 4.

Note that Maple can find the solution in one step:
> rsolve({c(n+1)= 2*c(n) + 115*c(n-1) + 364*c(n-2),c(0)=10, c(1)=7, c(2)=-17},c(k));

Find initial values for the sequence which would give a solution with one or 
two different eigenvalue powers in the final solution for c(k). 
Determine a link between these initial values and the eigenvectors of C.


This was the lab test from 2015, try to understand it and try the bits you are unsure 
of and ask me if you have any questions about it or anything else from past labs:


Unless otherwise specified, you can use any known command to answer.
You can check your answers after completion with any commands.

Do not look at the work other students are doing or communicate with anyone.
I recommend that you SAVE regularly in case your computer crashes.

Try to work sequentially, making sure your commands give the shown 
responses, do NOT erase or correct lines, just redo your calculation 
on the next line if you get an error. You will not lose marks for errors.

To write comments in your worksheet use the # symbol, do this to put 
your name and registration number at the top of your worksheet:
> # James Preen  20152613

Use
> #  the answer is ... because
if any of your lines need further explanation. 

Save your work in your H:\ directory with your name somewhere in the 
filename and then please email the file to me at james_preen@cbu.ca
Once I have confirmed receipt of the file you are free to leave.


Q0: (1 mark)
All students in every row and column of the room will have a different 
matrix to answer the questions on.

Using this diagram, identify which Matrix you will be using: 

A B D C

C E A B
=======    b       
B D C E    o
           a
E A B D    r
=======    d
D C E A

E B D C

         door

Students in the A position:
> M := Matrix([[3,2,6],[5,4,7],[6,6,2]]);
Students in the B position:
> M := Matrix([[3,4,3],[6,4,7],[4,6,4]]);
Students in the C position:
> M := Matrix([[7,3,5],[6,2,5],[7,5,3]]);
Students in the D position:
> M := Matrix([[6,5,4],[4,5,7],[4,3,2]]);
Students in the E position:
> M := Matrix([[2,2,3],[5,2,4],[2,4,5]]);


Q1: (10 marks)
Only using the command RowOperation to get the integer solution to M v = w
where
> w:=Matrix(3,1,[-6,-1,2]);
Check your result by matrix multiplication.

Q2: (9 marks)
Use any Maple commands to get the inverse of the transpose of M.

Now create a matrix E (by diagonalisation) with eigenvalues 2, 2 
and -6 which has the transpose of M as its eigenvector matrix. 

Compare your answer against what Maple gives 
> (v,P) := Eigenvectors(E);
and explain why P isn't exactly the same as the transpose of M,
demonstrating and identifying the relations between them.