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?

Inspired by Artin's Algebra

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