First, let us make an observation which will be used in
the proofs of the next two theorems:

**Observation:** *A square reduced row echelon matrix
with no zero rows is an identity matrix*.

**Remark:**

This is easily seen by considering ``at least'' and ``at most'' how many leading coefficients can occur on a square reduced row echelon matrix.

- is an invertible matrix.
- is row equivalent to the identity matrix .
- is a product of elementary matrices.

**Proof:**

By Theorem 3, is row equivalent to a reduced row echelon
matrix , i.e.
. If is invertible,
so is
. This
implies that
for some invertible . This further
implies that each row of must be non-zero, hence must be .

Next, if
,
. This shows that
is a product of elementary matrices.

Finally, since each elementary matrix is invertible, if is a product of elementary matrices, must be invertible.

**Q.E.D.**

**Corollary:** *If is an invertible matrix, then forming
a new matrix and performing row operations to to get
its reduced row echelon form. If we obtain an identity matrix on the left
hand side, then the right hand side is the inverse of .*

**Proof:**

Notice that , then .

**Q.E.D.**

**Corollary:** *If are matrices, then
is row equivalent to if and only if for some
invertible matrix .*