Matrix inversion: doing some housekeeping at elementary level
Any axiomatic treatment of the mathematical material should be preceded by a concrete, I would say naive, exploration.
What candidate would you suggest for the identity matrix?
Think simple, take a 2D case. Some students, even those who have taken matrix algebra, suggest I suggest analyzing the defining property , that is
Upon multiplication this becomes
This should be true for ANY Selecting and comparing elements in the upper left corner we see that Similarly, equating the elements in the lower right corner, we get The equation simplifies to
This implies Thus, This easily generalizes to higher dimensions:
Definition. The identity matrix is a square matrix with unities on the main diagonal and zeros outside it. The above exercise does not free us from the necessity of proving for all , which I leave to the reader.
Invertibility of matrices
Definition. Let be a square matrix. Its inverse, denoted satisfies by definition
We say that is invertible if its inverse exists.
Remark. (1) includes two equations: (right inverse) and (left inverse). We need both because of absence of commutativity. In fact, the existence of the right inverse implies the existence of the left inverse, and vice versa. There are many facts equivalent to invertibility.
Surprise #2. The condition is not sufficient for to be invertible.
Proof. Take Suppose it's inverse exists and denote it From the equation we have
The element in the lower right corner gives which is impossible.
How to use definition (1)
Rephrasing (1): whenever you have a matrix such that , you can conclude that
Example 1 (analog of , inverse of an inverse) . This is because by (1) satisfies
Example 2 (surprise #3, inverse of a product). If are invertible, then . To prove this, it suffices to check that the product inverts . We do this just for the right inverse:
(the identity matrix can be omitted)
(again by (1)) .