# 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

- \(A \overrightarrow{x} = \overrightarrow{b}\)
- 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?