# Finding a desired matrix

Find a matrix $A$ such that for three vectors $$v_1=(0,-1,1)^T, v_2=(2,1,1)^T,v_3=(-1,1,1)^T$$ $Av_1=1/2v_2$, $Av_2=1/2v_1$ and $Av_3=v_3$.

So we are looking for $A$ such that the matrix representation of $(Av_1,Av_2,Av_3)$ is $$\begin{pmatrix} 0 & 1/2 & 0 \\ 1/2 & 0 & 0 \\ 0 & 0 & 1\end{pmatrix}$$ But I m stuck on what to do next. Thanks

• Express the canonical vectors $e_1=(1,0,0)$, $e_2=(0,1,0)$ and $e_3=(0,0,1)$ as linear combination of the $v_i$, then compute their images $Ae_i$, and $A=(Ae_1\ Ae_2\ Ae_3)$. – Nihl Jul 21 '15 at 9:06

Note: $[Av_1\ Av_2\ Av_3]=A[v_1\ v_2\ v_3]$. Let $B=[v_1\ v_2\ v_3]$. By assumption, $AB=C$ where $C=\begin{bmatrix} 0& \frac{1}{2}& 0\\ \frac{1}{2}& 0& 0\\ 0& 0& 1\end{bmatrix}$. Thus $A=CB^{-1}$. $B^{-1}$ exists, by the way.