Start Maple from Start Menu/Program Files or using the icon in f:\math115.

Load the linear algebra progams with the command:
>    with(linalg):

Check the answers of the second test using these matrices:

>   A:=matrix(3,3,[-11,-4,12,18,3,-18,-8,-4,9]);
>   B:=matrix(4,4,[-1,-2,-1,-1,x,0,1,0,2,y,-2,1,0,-1,-2,0]);
>   C:=matrix(3,3,[1,-3,3,-1,1,-1,2,-2,3]);

Now try these eigen-questions:

Use determinant row and column operations to find the eigenvalues and eigenvectors 
of this 4x4 matrix:
>   G:=matrix([[0, 0, -1, -1], [-1, -1, -3, 1], [0, 0, 1, 0], [-1, 0, -1, 0]]);

	You may need to use 
> G1:=matadd(G,diag(-lambda,-lambda,-lambda,-lambda));
and
>   factor(c)
where c is the polynomial that results

Check your answer with 
>   eigenvals(G)
and 
>   eigenvects(G)
reading carefully the help pages which describe their format

Now use
>   evalm(G^5);
and then check your result by diagonalisation. 

Check that you get the same answer if you switch around the order of 
the eigenvalues in the diagonal matrix (and also the eigenvectors in P, of course).


Given H:=matrix([[0, 0, -1, -1], [-1, -1, -1, 1], [-1, 0, -1, 1], [0, 1, 1, 1]]);
Verify that [-4,2,3,5] is an eigenvector and so is [1,-2, 0,1]
Without using factor, work out the other two eigenvalues.

One more time, with this matrix 
J:=matrix([[-2, 6, 2], [-2, 5, 1], [-2, 4, 2]]);
Check that it has rank 3 but only two eigenvectors.

What are the eigenvalues of G^5 ? 
Try to predict the eigenvalues and eigenvectors of H^5 and J^5.



Form the matrix corresponding to the recurrence relation
  a(n+1)= -a(n)/6+49*a(n-1)/2+20*a(n-2)-200*a(n-3)/3
if we also have 
  a(0)=63, a(1)=48, a(2)=162, a(3)=-61
find the general solution for a(n) using the matrix method.

Check your answer using

>  rsolve({a(n+1)= -a(n)/6+49*a(n-1)/2+20*a(n-2)-200*a(n-3)/3,a(0)=63,a(1)=48,a(2)=162,a(3)=-61},a(n));


Using the polar bear/seal matrix
>  A:=matrix(2,2,[2/3,1/180,-12,6/5]);
diagonalise it and form
>  F:=multiply(P,E,inverse(P));
where E is the matrix with the nth powers of the eigenvalues

Find values of polar bears and seals to start with which will end up with
extinction of both after a number of years, and then find several values which
will end up with a stable population. What is the ratio of seals to bears?