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):

To stop you getting the same matrix as everyone else, use this command:

> Seed:=randomize():

Now create a 3 by 3 matrix by evaluating: 
> A:=RandomMatrix(3,3,generator=-8..7);

Maple can tell you the determinant with
> Determinant(A);

We can use the command SubMatrix to extract the cofactors from A
> C11:=SubMatrix(A,[2,3],[2,3]);

You can extract the value of the top left corner using
> a11:=A[1,1];

Create C12 and C13 similarly and combine the determinants to make
a Laplace Expansion by the first row and check that it gives
the same answer. Repeat by expanding by any column.

Do (legal) RowOperations on A 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 A and check that
it has the intended effect, 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 2A. Explain this by 
predicting what you should get for the determinant of (-3)A 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.

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 right value and sign.

We will now set one element of B to be an unknown:
> B[3,2]:=x:
We can now see the new B using
> B;
Find Determinant(B) and use these commands to check the answer which makes 
the new B singular:
> q:=solve(Determinant(B)=0,x);
> B1:=subs(x=q,B);
> Determinant(B1);

Now set B[4,4]:=y; and repeat this to find the determinant of B in terms
of x and y.
Note what polynomial is in the denominator of your new q and deduce the 
values of x and y which will make your new B be singular whatever y or x is.
Check your answer using "subs".

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 appear exactly where you expect in the inverse.

Check that you get the same result for the product of the determinant 
and inverse as you do for 
> Adjoint(E);

Set E[1,1]:=z; and note what happens in the adjoint and the inverse.
What is the adjoint if z is the value that makes E non-singular?