As usual start Maple and load all the linear algebra programs using 
> with(LinearAlgebra);
You can answer the questions in any order, please just write
comments with # to indicate what you are answering where.

We will use the command S:=SubMatrix(L,[1,2,3],[1,2,4]) to find the submatrix S made 
from matrix L by keeping only the 1st, 2nd and 3rd rows and the 1st, 2nd and 4th columns.
Note that the order of the numbers in the brackets is important.

Q1a
Define:
> A:=Matrix(3, 3, [[5, 6, 4], [3, 2, 6], [1, 2, 3]]);
Verify that 
> Determinant(A);
gives -32.
If we multiply a row of a matrix by k the resulting matrix will have a determinant k times that of the original. If we swap two rows of a matrix it will multiply the determinant by -1.
Use Transpose/RowOperation to make a matrix from A which has positive integers but has Determinant 16.
[1 mark]

Q1b
We choose row 1 of A and use SubMatrix and Determinant as follows to create the cofactors.
> f11:=SubMatrix(A,[2,3],[2,3]);
> f12:=SubMatrix(A,[2,3],[1,3]);
> f13:=SubMatrix(A,[2,3],[1,2]);
Note that I'm using frc as the name of the submatrix where r is a row number and c is a 
column number and in SubMatrix I am listing first the matrix, then the rows which I'm not deleting 
and then the columns which are not to be deleted. Note that r is the number missing from the first 
list and c is missing from the second list.
The order these rows and columns are given to Maple is important! 
Verify that 
> 5*Determinant(f11) - 6*Determinant(f12) + 4*Determinant(f13);
gives the correct determinant.
Repeat this procedure with new submatrices, such as f21, etc. and the appropriate numbers for 
row 2 and then for one of the rows of Transpose(A) to get -32 each time.
[2 marks]



Q2
Create a random 4x4 matrix M:
> randomize(Value(Time())): M := RandomMatrix(4, 4, generator = rand(2.. 7));
Maple can calculate the determinant directly:
> dM:=Determinant(M);
If the Determinant is zero, create a new M.

Q2a
Create a 1 somewhere in the matrix by using RowOperation and then add a multiple 
of the row with a 1 in it to another row to make a matrix M1 with a 0, and verify that
> Determinant(M1);
is the same as dM. Repeat for the rest of that column until there are zeros
above and below the 1 and let this matrix be MR. Show that the 
> Determinant(MR);
is still the same.
Now take 
> MT:=Transpose(MR);
and show its determinant is still the same value. 

Finally, use Pivot on any entry of MT not equal to 0 or in the 
same row or column as your original 1 was to form a matrix MP,
Verify the determinant of MP is still unchanged. 
Note: Please do not use ColumnOperation, use only Transpose and RowOperation.
[2 marks]

Q2b
Use SubMatrix to extract the 3 by 3 cofactor of MR corresponding to the row 
and column of the 1 you used and call it SR. 
Evaluate
> Determinant(SR);
and explain why it is the same as dM or not for your chosen pivot.
[1 mark]

Q2c
Now extract the 2 by 2 cofactor of the 4 by 4 matrix MP 
which is not in the same rows and columns as the two pivots, and call it SP.
Evaluate
> Determinant(SP);
and fully explain how and why it is related to dM.
[2 marks]


Q3
Define
> R:=<<0,1>|<-1,0>>;
and load the plots package
> with(plots);
Pick a pair of numbers x and y and set
> v1:=<x,y>
and evaluate 
> w1:=Multiply(R,v1);
Do this for half a dozen different points, v1, v2, ..., v6, and evaluate their products with R
and call them w1, w2, ..., w6.

Now plot them as follows:
> p1:=pointplot(<v1|v2|v3|v4|v5|v6>,symbol=solidcircle,color=red,symbolsize=20);
and then plot the products with R
> p2:=pointplot(<w1|w2|w3|w4|w5|w6>,symbol=solidcircle,color=green,symbolsize=20);
Using
> display([p1,p2]);
or otherwise, explain how your green dots are related to your red dots. You don't have
to say anything about eigenvectors or eigenvalues, I'll talk about that on Friday.	
[2 marks]

Once you have completed all of these activities, please save your worksheet then use
File / Export As and choose pdf. Now upload the file to Crowdmark.