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 ";"

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

Note that we use ":=" to assign a value to A. 

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.

We can add these matrices together using the + symbol
>    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 what you get when you add F to A.

We can take the transpose of any matrix with "Transpose",
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, then try to find the relationship 
between the product of A, transpose(B), B and transpose(A).

Create two random 3x3 matrices J and K. Repeat all of the earlier
questions for these two in place of A and B. 

Now multiply them first as J times K and then and K times J. 
Are the results the same matrix? 

Try to find a pair of matrices G and H for which this relationship gives 
the opposite result, so you have JK not equal to KJ 
and GH equal to HG or vice versa.