Finding the matrix $X$ in $AX$ 
Given that $A$ is a $3$ x $3$ matrix such that $$ A \begin{pmatrix}
        1 \\
        0 \\
        0 \\
        \end{pmatrix} = \begin{pmatrix}
        1 \\
        1 \\
        1 \\
        \end{pmatrix} $$ and $$ A \begin{pmatrix}
        0 \\
        0 \\
        1 \\
        \end{pmatrix} = \begin{pmatrix}
        0 \\
        0 \\
        1 \\
        \end{pmatrix} $$
Find a matrix $X$ such that 
$$A X = \begin{pmatrix}
        1 & 0 & 4 \\
        1 & 0 & 4 \\
        1 & 0 & 7 \\
        \end{pmatrix}$$

How do I approach this question? From the two matrix equations given above, I can find the first and the third columns, but I don't think I have the second column to find matrix $A$. Thanks. 

You can take$$X=\begin{pmatrix}1&0&4\\0&0&0\\0&0&3\end{pmatrix}.$$
And how do I know that? Note that$$X.\begin{pmatrix}1\\0\\0\end{pmatrix}=\begin{pmatrix}1\\0\\0\end{pmatrix}\implies AX\begin{pmatrix}1\\0\\0\end{pmatrix}=\begin{pmatrix}1\\1\\1\end{pmatrix},$$that$$X.\begin{pmatrix}0\\1\\0\end{pmatrix}=\begin{pmatrix}0\\0\\0\end{pmatrix}\implies AX.\begin{pmatrix}0\\1\\0\end{pmatrix}=\begin{pmatrix}0\\0\\0\end{pmatrix},$$and that$$X.\begin{pmatrix}0\\0\\1\end{pmatrix}=4\begin{pmatrix}1\\0\\0\end{pmatrix}+3\begin{pmatrix}0\\0\\1\end{pmatrix}\implies AX.\begin{pmatrix}0\\0\\1\end{pmatrix}=4\begin{pmatrix}1\\1\\1\end{pmatrix}+3\begin{pmatrix}0\\0\\1\end{pmatrix}=\begin{pmatrix}4\\4\\7\end{pmatrix}.$$
