Here are some issues that students, from both lectures, have brought up
in relation to homework 7.

1) WHEN DOING ROW OPERATIONS TO SOLVE  AX=B, FIND INV(A), OR LOOK
   FOR THE ROW REDUCED FORM OF  A MATRIX WHAT HAPPENS
   IF THERE IS A ZERO ON THE DIAGONAL?

   The essential goal of row operations is to get zeros  below the
   diagonal and have as many rows as possible, starting with the
   first, to have NOT zero.

   What to do if you hit a zero on the diagonal?
   Answer:
     a) You can interchange two rows of the augmented matrix.

     b) You can add a multiple of a row to the row that
        has a zero in the undesired place.

     c) If neither of these is possible then you have zeros
        from the diagonal down and the matrix is singular.


2) WHAT IS A ROW REDUCED FORM OF A MATRIX A?
    People have various definitions. All of the definitions
    have these properties:

      They are what the matrix A turns into when you do 
      the appropriate row  operations on the matrix or
      augmented matrix.

      They have nothing but zeros below the diagonal.

    More refined definitions of the reduced form of a matrix
    may demand that one do row operations (interchanging rows,
    adding multiples of one row to another, multiplying a
    row by a constant) until

       The first nonzero entry of every row is 1.

       Every row has more leading zeros than the previous
       row.

       The first nonzero entry of every row is the only nonzero 
       entry in its column. 

       For example, the MATLAB command rref(A) gives the following
       matrix as the row reduced form, this stronger kind
       of reduced form,  for some matrix A to be:

         1     0     0    -1
         0     0     1     2
         0     0     0     0
         0     0     0     0

      
    Whether you use the weaker form (further above), or the stronger form
    (just above), if there are any zeros on the diagonal of the reduced
    A then A is singular.