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

Remember to load the linear algebra package again
> with(LinearAlgebra);

Now define 
> A:=Matrix([[2, 1, -3, -1], [1, 0, -4, 3], [0, -3, -3, -1]]);
> b:=Vector([8, 16, -6]);
and make
> A0:=<A|b>;
as the system of equations we are planning to solve.

Instead of Row Operations this week we will use Pivot, which is just a set of 
row operations which uses one entry to make all others in that column zero.

Try
> A1:=Pivot(A0,1,4);
and find the Row Operations which give the same result.

Starting again with A0, choose appropriate entries to pivot around and create an 
equivalent of Reduced Row Echelon Form.
(Remember to use the RowOperation which multiplies rows by a number to make a 1)

Check your answer against Maple's:
> ReducedRowEchelonForm(A0);
Note the same ratios of numbers appear, even if the numbers are different.

Pivot by a different set of columns and see the same numbers appear again.

We can also ask Maple to solve the system of equations directly:
> q1:=LinearSolve(A,b,method='none',free=s);
Notice that without the "free" Maple will uses a complicated _t notation 
instead of our s. The index of s depends on which row is used by Maple, s[3] for us.

Check that this is indeed a solution for Av = b using:
> Multiply(A,q1);

Find the value of s[3] which will give you an actual solution <3,2,-1,3>, using
> subs(s[3]= fill in your choice here ,q1);
What are the other values of the solution when the first entry is 0?

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

Now create a random matrix
> F0:=RandomMatrix(4,4,generator=rand(1..3));

Use 
> Rank(F0);
and if it is 4, go up and press enter on the definition of F until you get one with
rank less than 4.
 alternatively, to save your time you can use a mini-program:
F0:=IdentityMatrix(4):while Rank(F0)=4 do F0:=RandomMatrix(4,4,generator=rand(1..3)); od:F0;

Pivot F0 until you have it in an equivalent of RREF and check your answer using
> LinearSolve(F0);


We have already seen how to Multiply matrices, see what you get with
> Multiply(A,F0);
and note that the number of rows and columns is as expected.

Note the error when you try to multiply them in the other order

Define
>    G:=RandomMatrix(2,2,generator=rand(-7..7));
>    H:=RandomMatrix(2,2,generator=rand(1..5));
and check that when you multiply G and H you will probably get a different 
answer depending on which order you multiply them in.

Use Maple's inverse function to create the inverse of G
>   Gi:=MatrixInverse(G);
and check that it has the form explained in class, and the property that
when multiplied by G on either side the 2x2 matrix identity arises.