### Inverse

##### 5. Properties of inverse

We have seen the relation between invertibility and the existence
of unique solution, which has immediate implication on the size of the matrix.
In addition to this, we have more properties for inverse transformations.

We start with a simple property.

**A** is invertible ⇒ **A**^{-1} is invertible,
and ((**A**)^{-1})^{-1} = **A**.

This is the consequence of the symmetry of the definition of inverse: **B** is inverse of **A**
⇔ **A** is inverse of **B**. Here is another simple property.

**A** and **B** are invertible ⇒
`AB` is invertible, and (`AB`)^{-1}
= **B**^{-1}**A**^{-1}.

It is important to note the order of multiplication.

A consequence of the two properties is that (**A**^{k})^{-1} =
(**A**^{-1})^{k} for any positive `k` and invertible **A**.
We denote (**AK**)^{-1} or (**A**^{-1})^{k}
by **AK** Combined with earlier notation (see the end of this
lecture), we have defined **AK** for any
invertible **A** and any integer `k`.

Proof (`AB`)(**B**^{-1}**A**^{-1}) =
**A**(`BB`^{-1})**A**^{-1} =
**A**(**I**)**A**^{-1} =
`AA`^{-1} = **I**. Similarly,
(**B**^{-1}**A**^{-1})(`AB`) = **I**.

The inverse of a matrix (or a linear transformation) is unique.

Proof Suppose **B** and **C** are inverses of **A**.
Then **B** = `IB` = (`CA`)**B**
= **C**(`AB`) = `CI` = **C**.

Finally, we assemble equivalent *invertibility criterion for square matrices*.
(see here for the proof)

For a square matrix **A**, the following are equivalent

**A** is invertible.
**AB** = **I** for some matrix **B**.
**BA** = **I** for some matrix **B**.
**Ax** = **b** has solutionss for any **b**.
- Solution of
**Ax** = **b** is unique.
- All rows of
**A** are pivot.
- All columns of
**A** are pivot.
**A** can be row operated to become **I**.

Note that in the original definition of inverse matrix, we need both the 2nd and the 3rd properties hold.
Now we see that *for a square matrix*, they are equivalent. The following are the same criterion stated for linear transformations.

For a linear transformation **T**: **R**^{n} → **R**^{n},
the following are equivalent.

**T** is invertible.
**TS** = `id` for some transformation **S**.
**ST** = `id` for some transformation **S**.
**T** is onto.
**T** is one-to-one.

If we do not know apriori the dimensions on both sides of **T**, we may use the following criterion for invertibility.

For a linear transformation **T**: **R**^{n} → **R**^{m}, the following are equivalent.

**T** is invertible.
**T** is onto and one-to-one.
`range`**T** = **R**^{m}, `kernel`**T** = {**0**}.

Proof By considering the matrices associated to linear transformations, this result then implies that **T** is invertible ⇔ **T** is onto and one-to-one. Conversely, suppose **T** is onto and one-to-one, then this result shows `m` ≤ `n` and this result shows `m` ≥ `n`. Thus `m` = `n`, and we may use the previous criterion to conclude that **T** is invertible. This completes the proof of the equivalence between the first two statements.

The equivalence between the second and the third statements follows from this definition and this result.