<s>[INST] Given a matrix, find a matrix that satisfies Let A be a matrix (3x4)
Prove that there does not exists a matrix X that satisfies
$$
        \begin{pmatrix}
        1 & 1 & 2 & -1 \\
        0 & 2 & 1 & 3 \\
        1 & 1 & 2 & -1 \\
        \end{pmatrix}X = \begin{pmatrix}
        1 & 1 & 1 \\
        0 & 2 & 0 \\
        2 & 1 & 1 \\
        \end{pmatrix}
$$
When I try to peform Gaussian elimination to get the reduced form of A, I always get a row of zeroes, e.g:
\begin{pmatrix}
        1 & 1 & 2 & -1 \\
        0 & 2 & 1 & 3 \\
        1 & 1 & 2 & -1 \\
        \end{pmatrix}
$$ R_3 - R_1 \to R_3 $$
I get
\begin{pmatrix}
        1 & 1 & 2 & -1 \\
        0 & 2 & 1 & 3 \\
        0 & 0 & 0 & 0 \\
        \end{pmatrix}
What can I conclude from the fact that I got a zeroes row? 
Does this help solving the problem?
 [/INST] The rows of the LHS will be given by the rows of $A$, multiplied by $X$.  Since the first and third rows of $A$ are the same, the first and third rows of the product will be the same.  Therefore the product cannot equal the RHS.
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