A ^{-1}A p= A ^{-1}p1 -2 0 11 2 0 1p= 1 -2 0 15 2

Ip= 1 -2 0 15 2

p= 1 -2 0 15 2= 1 2

This is (hopefully) the same answer you got for **p** by trial and
error a few pages ago.
If **A** is non-singular (has an inverse) and **A****p** = **q**,
then **p** = **A**^{-1}**q**.

The inverse of a non-singular square matrix is unique.
One way to see this is that there is only one
column matrix **p** that is the solution to
**A****p** = **q****A**^{-1}.

It might look like computing **A**^{-1} is a useful
thing to do.
In fact, **A**^{-1} is more useful in discussions
about matrices and transformations than it is in actual
practice.
Almost never do you really want to compute a matrix inverse.

For example, say that a column matrix **p** represents a
point in a computer graphic world.
The viewpoint changes, and the column matrix is transformed to
**q** = **A****p****A**^{-1}**q**.
But almost always there is an easier way to reverse
the transformation than to compute the inverse.