Don't forget to load the Linear Algebra package:
> with(LinearAlgebra);

Tell Maple about a new Matrix A:
> A:=<<9, 6, -4>|<-9, -6, 6>|<-3, -3, 5 >>;

Form the matrix from which we will get the eigenvalues:
> A1:=A-lambda*IdentityMatrix(3);

Q1 [4 marks]
From A1 do a row operation, a transpose and then another row operation to create two zeros.
Identify the eigenvalue that would result from a cofactor expansion using the zeros.

Now use SubMatrix to extract the 2x2 matrix S and evaluate
> factor(Determinant(S));
to identify the other two eigenvalues.

Repeat with a different row/transpose/row from A1 to identify a different submatrix 
determinant and check that the roots are the same as previously.

We can also get the system of equations to solve using this matrix:
> E1:=<subs(lambda=2,A1)|<0,0,0>>;
Pivot this matrix to create a row of zeros and find the solution, w.

Verify that w when multiplied by A on the left gives the expected result.
> Multiply(A,w);
> 2*w;


Q2 [2 marks]
If you were to repeat the RowOperation and Transpose procedure for 
> C:= <<-6, 2, -3>|<-13, 15, -15>|<-2, 8, -7>>; C1:=C-lambda*IdentityMatrix(3);
each way you try to do operations on C1 no easy cancellation occurs.
Remember you are NOT allowed to do row operations on C.

Use
> Eigenvalues(C);
to note what kind of factors you should get and try a few different row operationss 
to create one (no marks lost for not being able to, you get a mark for trying sensible things)

Maple can give the values and vectors directly:
> (v,P):=Eigenvectors(C);
Make a diagonal matrix from the values in v:
> D1:=DiagonalMatrix(v);

Verify that the product of P and D1 is the same as that of C and P.

Q3 [4 marks]
Create a random 3 by 3 matrix P with determinant -1 which will give your eigenvectors as follows:
> randomize(Value(Time())): P:=[1]: while Determinant(P)<>-1 do P:=RandomMatrix(3,3,generator=rand(-7..7)); od: P;

Make a diagonal matrix using two random integers which we will call a and b:
> a:=-3-2*(rand() mod 4): b:=2+(rand() mod 7): while b>=abs(a) do b:=2+(rand() mod 7); od: a; b;
> DA:=DiagonalMatrix(<a,b,b>);

Now make a matrix M using diagonalisation:
> M:=Multiply(Multiply(P,DA),MatrixInverse(P));
Similarly, find the value of the kth power of M using
> Dk:=DiagonalMatrix(<(a)^k,(b)^k,(b)^k>);
Call it Mtok. 
Check that k=1, k=0 and k=-1 give the appropriate matrices when put into Mtok.

We can multiply any vector such as 
> u:=<j, 60, 13> 
by the power of M
> uk := Multiply(Mtok, v0);
What value of j (likely a fraction) will give uk with no powers of a in?
Exactly how is this vector related to the eigenvectors from P?

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.