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

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

Remember that every Maple command must end with a semi-colon ";"
All the new commands listed are now known by Maple, we will use some.

We can create a 2x3 matrix by using the following command
>    A:=Matrix([[1,-1,2],[3,0,1]]);

Note that we use ":=" to assign a value to A. I recommend using
a capital letter, but Maple will not insist on that. All of the 
LinearAlgebra commands also begin with a capital letter, although
"matrix" is also defined, it will not always work properly.

Next we create a random matrix B, with integer entries between -4 and +5
>    B:=RandomMatrix(2,3,generator=rand(-4..5));

If you want to know more about a Maple command, type
>    ?RandomMatrix
to exit from the help page, close the subwindow, or use the 
Window menu to move between them.

As they are the same size we can add these matrices together using + 
>    C:=A+B;

D is a letter which is reserved for differentiation so we can't use it or
I which we saw in lab 0 was the square root of minus one. We can use
AA or DA if we want though, it is just simpler to keep single letters.

Check that C1:=B+A is the same as C and find E which is 3 times B 
minus 2 times A. Verify that alpha*(A-B) is equal to alpha*A - alpha*B.

Evaluate the scalar multiple
    F:=-1*A;
Check that you get the zero matrix when you add F to A.

We can take the transpose of any matrix with "Transpose" (capital T).

Try 
>  Transpose(A)
and see that the same numbers show up, just that rows 
become columns and vice versa

Check that Transpose(C) = Transpose(A) + Transpose(B)
and also predict and check what the transpose of E will be in terms
of the transposes of A and B.

We can, of course, multiply matrices, the command is "Multiply".
Try multiplying A and B and note the error given. Now verify the relationship like that given in class between the product of A, 
transpose(B), B and transpose(A) for our particular A and B.

Create a random 3x3 matrix J and define
> K:=Matrix([[-1, 5, 2], [-4, 2, -4], [1, 1, 2]]);

Now multiply them first as J times K and then and K times J. 
Are the results the same matrix? Check also that the square
of JK is not JJKK but JKJK and that Transpose(JK) is what you
expect too.

We can augment a column to our matrix K (but not make a dotted 
line unfortunately) using this command:
> K0:=< K | Vector([-1,-4,1]) >;

We can do the three different row operations as follows:
> K1:=RowOperation(K0,[1,3]);
will swap row 1 and row 3 of K and make it into a new matrix K1.
> K2:=RowOperation(K1,2,1/2);
will take this new K1 and multiply row 2 by 1/2 and make K2
> K3:=RowOperation(K2,[2,1],3);
will replace row 2 of K2 by itself plus 3 times row 1 to make K3

Note that this last operation will not give you the desired 0 in row 2
of column 1, so change this line to do so, then continue to reduce K
to reduced row echelon form.

Similarly, augment the same vector to your J and reduce.

Check your answer with
> ReducedRowEchelonForm(J0);