# Matrix Syllogism

Logic Level 4

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

1. $$A \overrightarrow{x} = \overrightarrow{b}$$
2. Multiplying both sides with $$L$$ on the left, $$L A \overrightarrow{x} = L \overrightarrow{b}$$
3. Hence, $$\overrightarrow{x} = L \overrightarrow{b}$$ satisfies $$1$$
4. Substituting $$x$$ back in $$1$$, $$A L \overrightarrow{b} = \overrightarrow{b}$$
5. 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?

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