As usual Maple from Start Menu/Program Files/ or using the icon
in f:\math115 and load all the linear algebra progams.

Now create a 3 by 3 matrix by evaluating: 
> A:=<<9, 6, -4>|<-9, -6, 6>|<-3, -3, 5 >>;

Use Maple to check the determinant of A and then form the matrix from which
we will get the eigenvalues:

> A1:=A-lambda*IdentityMatrix(3);

Use determinant row and column operations to create a situation from which
you could do a Laplace expansion and just have one 2x2 SubMatrix. Evaluate
the determinant of this submatrix and check it is the same when multiplied
in your laplace expansion as f:=Determinant(A1), which is the so-called
characteristic polynomial of A.
 
To manipulate polynomials you can do factor(f), expand((3-lambda)*Determinant(A3)) etc. Note that the determinant of A appears as the unit coefficient in f.
You can also plot f to see the locations of the eigenvalues using

> plot(f,lambda=1..4);

Find all eigenvectors using 

> v1:=LinearSolve(subs(lambda=3,A1),<0,0,0>);

and verify that all vectors satisfy Av = lambda v as expected.
What are the solutions if you try to substitute a non eigenvalue?

Try to repeat the above procedure for 
> B:= <<-6, 2, -3>|<-13, 15, -15>|<-2, 8, -7>>;
but note that each way you try to do operations on B1 no cancellation occurs.

Use
> Eigenvalues(B);
to note what kind of factors you should get and try to create one. Once you
have failed to create any zeros, just evaluate 

> db:=Determinant(B1); 

and check the eigenvalues using subs:

> subs(lambda=5, db);

Find one of the three eigenvectors by RowOperation and check for it in

> (v,P):=Eigenvectors(B);

Extract another eigenvector from P and check it has the expected eigenvalue.

Make the diagonal matrix of eigenvalues using

> D1:=DiagonalMatrix(v);

Check that the product and P and D1 is the same as that of B and P.

Make vi a vector of the ith powers of the eigenvalues and make Di 
a diagonal matrix of them, then multiply P and Di and the inverse of P
to get Bi.

Check that 
> subs(i=1,Bi);
and
> subs(i=0,Bi);
give the expected answers, and that
> subs(i=-1,Bi);
does too.

Check your answer with 
> MatrixPower(B,i);
The example from class was the following matrix:
> C:= <<-2, -3, -1>|<6, 9, 3>|<12, 18, 6>>;

Check our results using Eigenvectors and multiply C by itself to
get its square and cube.

Create a matrix which has the eigenvalues from w:
> randomise():w:=RandomVector(3,generator=rand(1..9));
> Dw:=DiagonalMatrix(w);
Choose a random invertible matrix for P and use diagonalisition to make J.
Check that the eigenvalues and eigenvectors of J are as you hoped.

Check these compound matrices have the expected values and vectors:
> Eigenvectors(k*J);
> Eigenvectors(Multiply(J,J));
> Eigenvectors(MatrixInverse(J));
> Eigenvectors(Transpose(J));

Make a random 4x4 matrix K and it should have horrible eigenvalues, check 
their decimal values using:
> evalf(Eigenvalues(K));

Form Kt:=K+Transpose(K); and note that all eigenvalues of Kt are real numbers.
(or would be if Maple didn't think they had a tiny imaginary part ;)

Plot the characteristic polynomials of K and Kt.