Consider a $m \times n$ matrix $A$ of coefficients, a vector $\overrightarrow{x}$ of unknowns and a vector $\overrightarrow{b}$ of solutions satisfying $A \overrightarrow{x} = \overrightarrow{b}$

You're given that there exists a left inverse $L$ of $A$ satisfying $L A = I_n$

Consider the following argument

- $A \overrightarrow{x} = \overrightarrow{b}$
- Multiplying both sides with $L$ on the left, $L A \overrightarrow{x} = L \overrightarrow{b}$
- Hence, $\overrightarrow{x} = L \overrightarrow{b}$ satisfies $1$
- Substituting $x$ back in $1$, $A L \overrightarrow{b} = \overrightarrow{b}$
- Hence, $L$ is a right inverse of $A$.

But that was not what we were originally given. Which is the first statement that went wrong?