Matrix Syllogism

Logic Level 4

Consider a m×nm \times n matrix AA of coefficients, a vector x\overrightarrow{x} of unknowns and a vector b\overrightarrow{b} of solutions satisfying Ax=bA \overrightarrow{x} = \overrightarrow{b}

You're given that there exists a left inverse LL of AA satisfying LA=In L A = I_n

Consider the following argument

  1. Ax=bA \overrightarrow{x} = \overrightarrow{b}
  2. Multiplying both sides with LL on the left, LAx=Lb L A \overrightarrow{x} = L \overrightarrow{b}
  3. Hence, x=Lb\overrightarrow{x} = L \overrightarrow{b} satisfies 11
  4. Substituting xx back in 11, ALb=bA L \overrightarrow{b} = \overrightarrow{b}
  5. Hence, LL is a right inverse of AA.

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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