Start Maple from the desktop, Start Menu/Program Files/ 
or using the icon in f:\math 1204

Load all of the linear algebra progams with the command:
>    with(LinearAlgebra);

Define
> K:=Matrix([[4, 4, -2], [1, 5, -4], [1, -3, 3]]);

We can augment a column to our matrix K (but not make a dotted 
line unfortunately) using the "|" symbol:
> K0:=< K | Matrix(3,1,[-4,-3,1]) >;

Use the command Pivot (use help to find out how it works)
to produce 2 zeros in column 1, then use RowOperation to
produce a row of all zeros. Use more RowOperations to 
produce one more 1 and 0 so you have an equivalent of 
Reduced Row Echelon Form.

Now use your brain and/or pen and paper to predict the 
particular and homogeneous solutions and input them using
> H:= < number, number, number >;
and 
> P:= < number, number, number >;
with the "number"s being your found values.

Now use Maple to check that you get the right answers
with the command "Multiply" as follows:
> Multiply(K,H);
> Multiply(K,P);
If it isn't wrong, try again until it works.

Start again with K0 but use a different column to start with 
and create a 1 using RowOperation, then Pivot around it.
Continue until you again have a different particular solution
and notice that the homogeneous solution is the same.

Verify that 
> Rank(K);
gives the expected answer explaining why in a comment.
You can add comments using 
> # it is why because

Notice how similar answers appear with the commands:
> ReducedRowEchelonForm(K0);
and 
> LinearSolve(K0);

Create a random 3x3 matrix using 
> M:=RandomMatrix(3,3,generator=rand(5..13));
First verify that Rank(M)=3, if not create a new M.

Create a 1 somewhere using RowOperations without fractions 
and then Pivot around it to make a matrix Ma. 
Now using more RowOperations starting with Ma to create 
another 1 without fractions in a different row and column.
You should now have exactly one digit left which isn't
in a row or column that contains a pivot point.

That number should be the result of 
> Determinant(M);
We will explain what this number is in class in a while...

Create a second 3x3 random matrix N and multiply it
by M, firstly with
> Multiply(M,N);
and then with
> Multiply(N,M);
and notice that these two are not the same, it is important
to remember that it matters with matrices which order they 
are being multiplied in.

Maple will refuse to multiply matrices of the wrong size,
create a 3x4 matrix and see that it can only be multiplied 
with K and M in one particular order and what error message
it gives when the matrices cannot be multiplied.

Create a matrix Q which does give the same answer when
multiplied on either side of M, but not N, or, if that
is too tricky, try to do it for 2x2 matrices.