Find Matrix $B$ s.t $Rowspan(A)=Rowspan(B)$ And $(1,1,1)\in Columnspan(B)$

Question

Let $A=\begin{pmatrix} 1 & 0 & 3 \\ 2 & 1 & 5 \\ -1 & 2 &-5 \end{pmatrix}$

Find a matrix $B$ such that $Rowspan(A)=Rowspan(B)$ and $\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\in Columnspan(B)$

So to find $Rowspan(A)$ I row reduced $\begin{pmatrix} 1 & 0 & 3 \\ 2 & 1 & 5 \\ -1 & 2 &-5 \end{pmatrix}$ to $\begin{pmatrix} 1 & 0 & 3 \\ 0 & 1 & -1 \\ 0 & 0 &0 \end{pmatrix}$ and because row operations do not change the row span so $B$ should span the same space of $CF(A)$ and $\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\in Columnspan(B)$

But where should I start?

Answer 1

From here

$$A^{RREF}=\begin{pmatrix} 1 & 0 & 3 \\ 0 & 1 & -1 \\ 0 & 0 &0 \end{pmatrix}$$

you are done, indeed just consider

$$B=\begin{pmatrix} 1 & 0 & 3 \\ 0 & 1 & -1 \\ 1 & 0 &3 \end{pmatrix}$$

and note that $$B\cdot (1,1,0)=(1,1,1)$$