One difficulty that students often encounter is how to get started in proving a theorem like this. The difficulty is certainly not in the theorem’s complexity. It’s too terse! Before actually starting the proof, we rephrase the theorem so that the implication it states is clear.
The same problem is encountered here as in the previous theorem. We will leave it to the reader to rephrase this theorem. The proof is also left to the reader to write out in detail. Here is a hint: If
and
are both inverses of
then you can prove that
If you have difficulty with this proof, note that we have already proven it in a concrete setting in Chapter 5.
As mentioned above, the significance of
Theorem 11.3.3 is that we can refer to
the inverse of an element without ambiguity. The notation for the inverse of
is usually
(note the exception below).
The next theorem gives us a formula for the inverse of
This formula should be familiar. In Chapter 5 we saw that if
and
are invertible matrices, then
We prove the theorem only for since the second statement is proven identically.
By the cancellation law, we can conclude that
If and are two solutions of the equation then and, by the cancellation law, This verifies that is the only solution of