### Inverse

Exercise Prove that if row operations change [`A B`] to
[`I X`], then **X** = **A**^{-1}**B**.

Answer Since the same row operations change **A** to **I**,
by this criterion, we see that **A** is invertible.

Let the `i`-th columns of **B** and **X** be **b**_{i}
and **x**_{i}. Then the same operations change [**A b**_{i}] to
[**I x**_{i}].
Equivalently, `Ax` = **b**_{i} and `Ix` = **x**_{i}
have the same solutions. This implies **Ax**_{i} = **b**_{i}.
Then by this formula, `AX` =
[`Ax`_{1} `Ax`_{2} ... **Ax**_{n}] =
[**b**_{1} **b**_{2} ... **b**_{n}] =
**B**. Since **A** is invertible, we get
**X** = **A**^{-1}**B**.