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.

Randomise your session using 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 random matrix until it is.

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 to 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, and is the same as 
> Adjoint(H);


We can use the command SubMatrix to extract the cofactor matrices from H
> C12:=SubMatrix(H,[2,3],[1,3]);
We are telling Maple to consider from H rows 2 and 3 and columns 1 and 3.
Note that it is rows then columns, and to list the rows/columns in increasing order.

Identify the position in Ha and the sign that 
> dc12:=Determinant(C12);
corresponds to.

You can extract the corresponding value of the H matrix entry using
> h12:=H[1,2];

Repeat to extract the 2x2 matrices for C11 and C13 and combine their 
determinants to make a Laplace Expansion by the first row:
> h11*dc11 - h12*dc12 + h13*dc13;
check that it gives the same answer as dh.

Now re-expand in the same way by the second column of H and check that it works again.


Do type 1 RowOperations on H to produce two zeros in a row or column
and then take the determinant and check your answer.

Use the RowOperation which switches two rows of H and check that
it makes the new determinant negative.

Use the RowOperation which multiplies a row by a number and noticing how 
that changes the determinant, then find the determinants of 2*H and (-3)*H
and compare them to dh.


Form a new random 3x3 matrix E and set 
> E[3,1]:=x;
Note that to see the whole matrix E you should type the following:
> E;

Use the commands Determinant and solve (not Solve, it doesn't exist) to find 
which value of x would make E singular.
> solve(Determinant(E)=0,x);

Now set 
> E[2,3]:=y;
check that E has both an x and a y in it, and then use solve
to find out which value of y would make E guaranteed to be non-singular.
Check the determinant by using
> subs(y=your value, Determinant(E));

Repeat with a similar command for the value of x which will guarantee an inverse.

Check that the inverse of HE is the product of the inverses of E and H, and that the
determinant of EH is the product of the two determinants in either order.