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

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

If you end a Maple command with a colon ":" it doesn't show any output

We define a matrix:
>  E:=Matrix([[1,3,-1,5],[-2,1,-5,4],[0,-1,1,-2]]);

Verify that E has rank 2 by typing
>  Rank(E);
check that the RREF supports this claim using:
>  ReducedRowEchelonForm(E);

Check that the rank of the transpose of E is also 2 by looking
at its RREF and check with the command Rank.

Choose a random 4x1 matrix and multiply E by it to get a matrix B.
Form E1 by augmenting B to E and take the RREF of E1.

Maple can solve this system of equations for us:
> X:=LinearSolve(E, B, 'free=alpha');
Check that the output matches what you would have got from the RREF.
Identify the values of the alphas which correspond to your initial matrix.
Use the command subs to check your substitution:
> X1:=subs(alpha[3]=1,X);

Create a random 4x4 matrix F with entries between -1 and 2 and make sure that
> Rank(F) = 4. 
Solve the matrix equation 
       F <x1, x2, x3, x4> = <5,-1,3,2>;
using RowOperation.

Maple can find inverses of square matrices for you automatically:
>    MatrixInverse(F);

Verify that the product of the inverse of F and <5,-1,3,2> gives you
the expected answer.

Note that when you ask Maple for the inverse of E (the non-square matrix
from earlier) it gives you an answer, but it is using advanced matrix
theory which we will not cover in this course.

Form the 3x3 zeroes matrix and see what Maple says when it has no inverse.
>    Z:=Matrix(3,3,0);

Generate several random 2x2 matrices and find two whose inverses do exist.

Multiply these inverses and check that the relation for the
inverse of a product is the product of the inverses in the other order.
Check this relation still holds for three of your matrices multiplied.

Check that there is no easy relation between the inverse of 
(A+B) and the inverses of A and B

Find two matrices V and W such that both V and W have inverses but (V+W) doesn't

We can also use row operations to find inverses, if we take the matrix
>   J:=Matrix([[-1,-2,2],[1,-2,-1],[-1,1,1]]);
and augment to it the 3x3 identity matrix which we can get using 
>   Ij:=DiagonalMatrix([1,1,1]); 
and
>   JI:=<J|Ij>;

Use RowOperation to get the inverse then verify your solution with RREF.

Verify that the transpose of the inverse of J is the inverse of its transpose.

Find MatrixInverse(F) and Determinant(F) and note the appearance of the
latter in the former. Verify that the laplace expansion of a random
row works by forming the submatrices using commands like:
> F1:=DeleteRow(DeleteColumn(F,1),1);