---
kind: statement
msc:
- 15A09
name: Properties of invertible matrices
role: proposition
sources:
- locator: src/section3-1.xml:806
  url: https://github.com/davidaustinm/ula/blob/b6ea263ef23703072b7007485bb7a2d89c5f399b/src/section3-1.xml#L806
tag: KAJZSCZQ5C
uses:
- 0EB9N0S33C
- 006BHFWMAP
- 0WF5AQYTW6
---

## Statement

Let $A$ be an $n \times n$ matrix. The following four facts hold.

1. The matrix $A$ is [[invertible|0EB9N0S33C]] if and only if $A$ is [[row equivalent|006BHFWMAP]] to the identity matrix $I$.
2. If $A$ is invertible, then for each vector $\mathbf b$ the equation $A\mathbf x = \mathbf b$ has the solution $\mathbf x = A^{-1}\mathbf b$.
3. If $A$ is invertible, then row reducing the augmented matrix $[A \mid I]$ gives its [[reduced row echelon form|0WF5AQYTW6]] $[I \mid A^{-1}]$. This is a way to compute $A^{-1}$.
4. If $A$ and $B$ are invertible $n \times n$ matrices, then $AB$ is invertible and $(AB)^{-1} = B^{-1}A^{-1}$.

## Remarks

Fact 2 follows by multiplying both sides of $A\mathbf x = \mathbf b$ on the left by $A^{-1}$.

Fact 4 is checked by the product $(AB)(B^{-1}A^{-1}) = I$.