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.

We can define matrices by columns as follows:
>  F:=<<-3, 5, -1, -1>|<4, -4, -2, -5>|< 5, 2, 4, -5>|< 11, 0, -4, -22>>;
>  w:=<-8,3,-5,4>;
and augment F and w using the | notation too:
>  F1:=< F | w >;

Use RowOperation to get F1 into Reduced Row Echelon Form in standard form 
starting at the top left with a 1 and call it your final answer Ff.

We can use a new command to solve from here:
> v:=BackwardSubstitute(Ff);
(this command only works with a matrix in RREF form)
Note the use of _t1 to mark a free variable.

Multiply F and v and note that you get w.

Use this command to substitute a random value for the free variable
> vr:=subs(_t[1]=rand(-3..3)(),v);
Check that multiplying F and vr gives you the expected answer too.

Now augment the zero vector to F to get Fz and check you get the expected 
result using 
> Fzf:=ReducedRowEchelonForm(Fz);
And check that BackwardSubstitute(Fzf) is what you expect too.

This command will tell you the number of non-zero rows you got in RREF of F:
> Rank(F);

Create a 3 by 3 matrix by evaluating: 
> G:=Multiply(<<1,2,-1>>,RandomMatrix(1,3,generator=-5..6));
and notice that each row is a simple multiple of the first.
Think about what you expect the rank to be, and then check it using Rank.

Augment <-3,4,3> to G and note what the RREF of this augmented matrix
is and what error message BackwardSubstitute gives. Change <2,4,3>
to something else which will have a solution u and verify it by multiplying.

Identify the two homogeneous solutions from u and the particular
solution, create them individually using the comma notation and 
multiply each by G to verify they all have the appropriate properties.


Create a random 3x3 matrix H with entries between -4 and 4 and ensure that
Rank(H) is 3, if not create a new matrix.

Multiply H by itself and check the rank of this new matrix. 

Choose a new 3x3 matrix J with rank 2 and evaluate the rank of J^2

What are the ranks of JH, KH and H+J? Do you suspect any pattern?
Find a 3x3 matrix L with Rank 3 such that L+J has a different rank to K+J.
Find a 3x3 matrix M such that MJ has a different rank to KJ.