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.