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If I have two matrices

$ A = \begin{bmatrix} 1 & 1 & 2 \\ 1 & 2 & 3 \\ 0 & -3 & -2 \\ \end{bmatrix} $, $ AX = \begin{bmatrix} 2 & -2 \\ 1 & -1 \\ 3 & -3 \\ \end{bmatrix} = B$

how can this be [A|B] ~ [I|X]? I don't understand the logistics behind it. I understand that $AA^{-1} = I$ but I don't know how to go from there.

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The solution is what you stated: $$ AX=B\Rightarrow A^{-1}AX=A^{-1}B\Rightarrow(A^{-1}A)X=A^{-1}B, $$ hence: $$ X=A^{-1}B. $$

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