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

Load all of the linear algebra progams with the command:
>    with(LinearAlgebra):
A colon instead of a semi-colon suppresses the output of all the names.

To stop you getting the same matrix as everyone else, use this command:
> Seed:=randomize():

Create a random 3x3 matrix H 
> H:=RandomMatrix(3,3,generator=-4..5);
ensure that Rank(H) is 3, if not create a new matrix.
Using
> I3:=IdentityMatrix(3);
form the augmented matrix
> H1:=< H | I3 >;
and use row operations to transfom H into I3, trying to keep your matrix as integers as long as possible.

Maple can calculate the inverse of any matrix too, of course:
> Hm:=MatrixInverse(H);
Check that the product of Hm and H is I3 as well as the product of H and Hm.

Check that the common denominator of the inverse is the determinant with
> dh:=Determinant(H);
by checking that Ha:=dh*Hm is an matrix made of integers.


We can use the command SubMatrix to extract the cofactors from H
> C11:=SubMatrix(H,[2,3],[2,3]);
We are telling Maple to only consider rows 2 and 3 and columns 2 and 3

You can extract the value of the top left corner using
> h11:=H[1,1];

Repeat for C12 and C13 and combine the determinants to make
a Laplace Expansion by the first row and check that it gives
the same answer. Now re-expand by the second column of H.

Do (legal) RowOperations on H to produce two zeros in a row or column
and then do a Laplace expansion and check your answer.

Use the RowOperation which switches two rows of H and check that
it makes the determinant negative, and do the same using ColumnOperation.

Use the RowOperation which multiplies a row by a number and see how 
that changes it, and then find the determinant of 2*H. Explain this by 
predicting what you should get for the determinant of (-3)*H and checking.

Now create a random 4x4 matrix B and check that your idea for the
determinant of a scalar product of B still holds.
Perform a Laplace Expansion on any row or column of B.

Use legal ColumnOperations to manipulate B so that one row has 3 zeroes 
in and then extract the submatrix corresponding to it and evaluate
its determinant and check that you get the correct value and sign.

Create E as a random 3x3 matrix and check its inverse using MatrixInverse.
Get the determinant of E and the determinants of all 9 2x2 submatrices and
verify that they all appear in the inverse and note the pattern.

Check that the inverse of HE is the product of the inverses of E and H