As usual run Maple and load all the linear algebra progams.

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 with two zeros as coefficients. 
ColumnOperation works in exactly the same way as RowOperation.

We can find the polynomial which contains the eigenvalues using:
> pA:=Determinant(A1); 
which is the so-called characteristic polynomial of A.
Note that the coefficient of pA without any lambda in is the determinant of A.

To manipulate polynomials you can use factor, expand, sort, etc. 
> factor(pA);

You can also plot pA to see the locations of the eigenvalues using

> plot(pA,lambda=1.5..4);

Find the eigenvectors belonging to the eigenvalue 2 using 

> v1:=LinearSolve(subs(lambda=2,A1),<0,0,0>,free=s);

and verify that all vectors satisfy Av = lambda v as expected.

Find two eigenvectors of the eigenvalue 3 similarly and check.
What are the solutions if you try to substitute a non eigenvalue?

Try to repeat the above procedure for 
> C:= <<-6, 2, -3>|<-13, 15, -15>|<-2, 8, -7>>;
> C1:=C-lambda*IdentityMatrix(3);
but note that each way you try to do operations on C1 no easy cancellation occurs.

Use
> Eigenvalues(C);
to note what kind of factors you should get and try to create one. If after 5 minutes
if you have failed to create two zeros in a row, just evaluate 
> db:=factor(Determinant(C1)); 
to reassure yourself the answers Maple gave are indeed correct.

Find one of the three eigenvectors in C using RowOperation or Pivot and check for 
it or a multiple of it in the columns of P in

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

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

Make a diagonal matrix of eigenvalues using

> D1:=DiagonalMatrix(v);

Check that P and D1 multiplied is the same as C and P multiplied in that order.

Check these compound matrices have the expected values and vectors:
> Eigenvalues(k*C);
> Eigenvalues(MatrixInverse(C));
> Eigenvalues(Transpose(C));

Think about what these will always work in this way.

Find the eigenvalues of this matrix using determinant row and column operations.

> E := Matrix([[3, -8, -3, 7], [2, -5, -3, 5], [2, -4, 2, -1], [2, -4, 0, 1]]);

To extract you can use the commands DeleteRow and DeleteColumn, or SubMatrix.
(SubMatrix works by specifying which rows and columns you wish to keep)

Find two eigenvectors of the eigenvalue of multiplicity 2 by pivoting.

Please randomise again:
> Seed:=randomize():

Create a random 4x4 matrix B using this command:
> B:=RandomMatrix(4,4,generator=-7..6);
We will now set three elements of B to be unknowns in terms of x:
> B[3,2]:=x: B[1,3]:=x-2: B[4,1]:=2*x+3:
We can now see the new B using
> evalm(B);

Find Determinant(B) using a co-factor expansion utilising "SubMatrix"
to get the four different 3x3 determinants required.

Use the following commands to find an answer which makes 
the new B singular:
> q:=evalf(solve(Determinant(B)=0,x));
Note that you may have some complex numbers as solutions.
Assuming you have a real solution in position 1 of q, use:

> B1:=subs(x=q[1],B);
> Determinant(B1);
Note: do not set "x:=q;" as then x will always have this value. 
Use x:='x'; to restore it as a variable.