Determining the matrix of the form AX

Question

I want to solve problem b) Here is my solution:

I followed the hint and found the column space of matrix $A$ $$\begin{bmatrix}1 & 0 \\-1 & 1\\0 &-1\end{bmatrix}$$

Equation $Ax = b$ has solution only if $b$ is in column space(we can express b as a linear combination of the columns of matrix $A$).

But i don't know what to do next. How can I find an answer on question b)?

Edited: the answer is

Answer 1

Suppose a matrix $B$ is of the form of $B=AX$, then

$$B=A \begin{bmatrix}x_1 & x_2 & x_3 \end{bmatrix}= \begin{bmatrix}Ax_1 & Ax_2 & Ax_3 \end{bmatrix}$$

Hence the columns of $B$ must be in the column space of $A$.

Note that

$$span \left( \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 1 \\ -1 \end{bmatrix} \right) = span\left( \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} , \begin{bmatrix} 0 \\ 1 \\ -1 \end{bmatrix} \right)=\left\{ a\begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} +d \begin{bmatrix} 0 \\ 1 \\ -1 \end{bmatrix}: a, d \in \mathbb{R}\right\}$$