TITLE: Using the Left-Inverse to "Solve" an Impossible System of Equations
QUESTION [28 upvotes]: I was working with the following system of equations:
$$\begin{split}
    \begin{bmatrix}
    4 & 0\\
    0 & 5\\
    0 & 0\\
    \end{bmatrix}
    \begin{bmatrix}
    x_1\\
    x_2\\
    \end{bmatrix}
     & = 
    \begin{bmatrix}
    1\\
    1\\
    1\\
    \end{bmatrix}
    \end{split}
$$
Clearly, this has no solution on account of the last rows in the coefficient and solution matrices. 
However, multiplying by the left-inverse of the coefficient matrix seems to imply a solution:
$$\begin{split}
    \begin{bmatrix}
    \frac{1}{4} & 0 & b_{13}\\
    0 & \frac{1}{5} & b_{23}\\
    \end{bmatrix}
    \begin{bmatrix}
    4 & 0\\
    0 & 5\\
    0 & 0\\
    \end{bmatrix}
    \begin{bmatrix}
    x_1\\
    x_2\\
    \end{bmatrix}
     & = 
    \begin{bmatrix}
    \frac{1}{4} & 0 & b_{13}\\
    0 & \frac{1}{5} & b_{23}\\
    \end{bmatrix}
    \begin{bmatrix}
    1\\
    1\\
    1\\
    \end{bmatrix} \\
    \begin{bmatrix}
    x_1\\
    x_2\\
    \end{bmatrix}
     & = 
    \begin{bmatrix}
    \frac{1}{4} + b_{13}\\
    \frac{1}{5} + b_{23}\\
    \end{bmatrix}
    \end{split}
$$
Where $b_{13}$ and $b_{23}$ can be any number. As stated above, there is no solution to this system, so any solution obtained by the above method is wrong.
What I am not understanding is why, after applying the rules of matrix multiplication, does it seem possible that there is a solution? Did I make a mistake somewhere in the process, and/or is there some fundamental nuance of linear algebra that I'm missing here?

REPLY [2 votes]: One of the principles of classical logic is that if we accept any false statement as true, then it is possible to prove anything ("paradox of entailment"). 
Your initial equation implies 0*x1 + 0*x2 = 1, i.e. 0 = 1. By manipulating the two sides of this "equation" we can then get any mathematical result we like.