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


                      door A B C D 
              C D A B      
              =======      
              A B C D      C D A B
                           =======
              C D A B      A B C D
              =======
              A B C D      C D A B
                           =======
              C D A B      A B C D
              =======      
              A B C D      
   entrance B184           C D A B  entrance B186

All students in every row and column will have a different 
(but equally hard) matrix to answer the questions on.

Students in the A position:
M := Matrix([[1, 1, 1, 3], [4, 1, 3, 3], [3, 1, 1, 2], [3, 1, 1, 2]]);
Students in the B position:
M := Matrix([[4, 4, 2, 4], [2, 2, 1, 2], [3, 4, 1, 1], [1, 4, 3, 1]]);
Students in the C position:
M := Matrix([[3, 2, 2, 2], [2, 2, 2, 2], [2, 3, 3, 3], [4, 4, 1, 2]]);
Students in the D position:
M := Matrix([[1, 2, 4, 1], [1, 3, 3, 1], [3, 4, 1, 1], [1, 4, 2, 1]]);

Do not look at the work other students are doing or talk with them.
 
Unless otherwise specified, you can use any known command to answer.
You can check your answers after completion with any commands.

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.

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  26262626

Use
> #  the answer is ... because
if any of your lines needs 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
I recommend that you save regularly in case your computer crashes
Once I have confirmed receipt of the file you are free to leave.

Start your session by loading the LinearAlgebra package as usual.

Q1: (10 marks)
Using only RowOperation, find an all-integer solution vector for M times v = 0. 
Check your answer using Multiply with both your original matrix and your RREF.

Q2: (5 marks)
Set two positions in M to the following letters:
> M[2,1]:=x; M[3,3]:=y;
Take the determinant and hence explain which values of x or y 
guarantee that M will be singular. Why can M never be guaranteed 
non singular by a choice of x?

Q3: (5 marks)
Use Maple to create a random 2x2 matrix P;
> Seed:=randomize(): P:=RandomMatrix(2,2,generator=rand(1..5));
Check if P has rank 2. If it isn't, try again until it is.

Use P to create a 2x2 matrix A with eigenvalues 1 and -1 by diagonalisation:
> A:=Multiply(P,Multiply(DiagonalMatrix(<1,-1>),MatrixInverse(P)));

Use a similar command to find the general expression for A to 
the power k and give the matrix which is A to the power 14. 
Explain why this happened and how it is related to the inverse of A.