### Inverse

Exercise Let **A** and **B** be square matrices.
Prove that **A** and **B** are invertible ⇔ `AB` is invertible.
Please also explain why the square condition is necessary.

Answer In the lecture, we have already shown the direction ⇒.

Now we assume `AB` is invertible and try to prove **A** and **B** are invertible.
Let **C** be the inverse of `AB`. Then we have `ABC` = **I** and
`CAB` = **I**. By this criterion,
the fact that **A**(`BC`) = **I** and **A** is square implies
**A** is invertible. Similarly, the fact that (`CA`)**B** = **I**
and **B** is square implies **B** is invertible. This completes the proof for the direction ⇐.

The square condition is necessary because nonsquare matrices are not invertible. For a concrete example, consider

**A** = [ |
1 |
-2 |
], **B** = [ |
3 |
] |

1 |

We have `AB` = **I**, but **A** and **B** are not invertible.