3.3-8
=====
A matrix is in reduced row echelon form <==> the corresponding homogeneous
system of equations is a reduced echelon system. There is a simple criterion
for testing if a given matrix B is in reduced row echelon form:

   TEST: Each nonzero row of B has a leading one, and the leading one
         sits in a column of the identity. Finally, these identity
         columns appear in natural order as consecutive initial columns
         of the identity.

The textbook reports the answer as the 3x3 identity matrix.
Details expected include a frame sequence that starts with matrix A and 
ends with the identity. Document all steps with detailed information in
the form swap(s,t), combo(s,t,c), mult(t,m).


3.3-18
======
The textbook reports the correct answer as a 3x5 identity matrix B.
Details expected include a frame sequence that starts with matrix A and 
ends with matrix B. Document all steps with detailed information in
the form swap(s,t), combo(s,t,c), mult(t,m).

The matrix B is a reduced row echelon form, because the nonzero rows are
rows 1 and 2, and each contains a leading one, which sits in a column of
the 3x3 identity matrix I. These columns are the first two columns of I
in natural order.

3.3-34
======
The problem is to list all possible 3x3 matrices A in reduced echelon form.
No answer is given in the book. 

Start by letting A=[[a,b,c],[d,e,f],[g,h,i]]. Look at the number of leading ones,
called the rank(A).

Case 1. rank(A)=3

 Then A=[[1,0,0],[0,1,0],[0,0,1]; there is only one reduced form.

Case 2. rank(A)=2

 Then A has exactly one zero row, which must be the last one, so

     A=[[a,b,c],[d,e,f],[0,0,0]]
 
 To be in reduced form, a=1, b=0, d=0, e=1:

     A=[[1,0,c],[0,1,f],[0,0,0]]

 Symbols c,f can be anything.

Case 3. rank(A)=1

 Argue that

     A=[[1,b,c],[0,0,0],[0,0,0]]

 where b,c can be anything.