Remember from Section 2.1 (Week 4 Tuesday) that we had three formulas
for matrix multiplication.


In section 2.2 the key concepts are:  a square matrix is either
invertible, or singular.  For 2x2 matrices you find a formula
for the inverse on page 105.
For larger matrices, the best way to compute the inverse of A
is to row-reduce (A | I) to (I | ...)
(if A can not be row-reduced to I then A is not invertible).
After that you can read of the inverse of A   (see page 110).

In section 2.3 we start with a square matrix A, and then Theorem 8
gives numerous statements, all of which are equivalent to saying
that A is invertible.
So if A is singular, then all statements in Theorem 8 are false,
and if A is invertible, then all of them are true.

Notice that in order to know what items e,f,h in Theorem 8 say, you
need to know (memorize!) the definitions from Chapter 1.  Also, you
need to know that item g is the same as saying that A is surjective (onto).


Exercises:
2.2 1-4  : memorize formula in Theorem 4.
    5-6  : do one of these
      8  : Was explained this in class, can you reproduce the argument?
           Hint: another way to do this is to say this: we are told
           that A is invertible, so A^(-1) exists.  Now take the
           equation AD = I and multiply both sides on the left by A^(-1).
           What do you find?
     11 :  Same argument as in 8.
     12 :  This was also explained in class.  Remember if we can row-reduce
           (A | B) to (I | ...) then we can read off the solution of AX = B.
     13 :  Same argument as in 8.
     14 :  This time multiply both sides on the right by D^(-1)
           and you find (B-C)D D^(-1) = 0 D^(-1). Then simplify.
     Go through 15-18.  Hint for 18: multiply both sides on the left
     by P^(-1) and on the right by P.
     21-24 : For this, you need to understand the definitions from Chapter 1
             quite well (you will also find the answer to 21-24 in
             the next section, section 2.3).
   29-33 : memorize algorithm on page 110.

2.2 Turn in: Ex 8, 9, 10, 11.


2.3
  Practice problem 1. After a single row-reduction R3 <- R3 - R1 you
  can already stop, you'll get a zero-row, so there can't be 3 pivots
  in RREF, so RREF can't be I, so A can't be invertible.
  Can you find other items in Theorem 8 that will allow you to decide
  quickly that A is not invertible? 
  More on this is found in Ex 1-10.  In those, do as little as possible.
  For instance, in Ex 8 you don't need to do any work at all, it is a 4x4
  matrix, in REF, with 4 pivots  (item (c) in Theorem 8).  Ex 3 is similar
  except that that one is in REF after you take the transpose (item (l)).
  So the point of Ex 1-8 is not to do a lot of computations, it is to
  learn shortcuts.

2.3 Turn in 11, 12, 13.

   (answers for 11, 12 are very short true/false but to find the answer
    you may have to go back and read the theorem).

   Note:
   Ex 14 is basically the same as Ex 13 after you take the transpose.
   Ex 15 : hint: Theorem 8 item (e).
   Read the exercises after this to test your understanding (remember
   that we did Ex 28 in class?)

2.4 We did part of this section in class, one more theorem to do to
    finish this section.  I'll assign exercises for 2.4 and 2.5 next time.