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

Remember to load the linear algebra package again, you can use
> with(LinearAlgebra):
A colon instead of a semi-colon suppresses the output of all the names.

Now recall that we defined 
> A:=Matrix([[2, 1, -3, -1], [1, 0, -4, 3], [0, -3, -3, -1]]);
> b:=Vector([8, 16, -6]);
as the system of equations we were planning to solve.

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

Extract the particular and homogeneous solutions using these commands and check 
that they multiply with A correctly:
> qp:=subs(s[3]=0,q1);
> qh:=subs(s[3]=1,q1-qp);

Find the value of s[3] which will give you a particular 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..2));

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 and the rank using
> ReducedRowEchelonForm(F0);


To Multiply matrices we use this command (not *), 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.

Also check that 
>    Gt:=Transpose(G);
and
>    Ht:=Transpose(H);
multiply as expected to give 
>    Transpose(Multiply(H,G));
only in this order.

Create a random 3x3 matrix J 
> J:=RandomMatrix(3,3,generator=-4..5);
ensure that Rank(J) is 3, if not create a new random matrix until it is.

Using
> I3:=IdentityMatrix(3);
form the augmented matrix
> J1:=< J | I3 >;
and use row operations to transfom J into I3, trying to keep your matrix 
as integers as long as possible.

Maple can calculate the inverse of any matrix too, of course:
> Jm:=MatrixInverse(J);


Now check the solutions for test 1 using Maple or
continue with lab1 if you haven't already finished it.