Properties of invertible matrices

Statement

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

1. The matrix $A$ is invertible if and only if $A$ is row equivalent 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 $[I \mid A^{-1}]$. This is a way to compute $A^{-1}$.
4. If $A$ and $B$ are invertible $n \times n$ matrices, then $AB$ is invertible and $(AB)^{-1} = B^{-1}A^{-1}$.