Answer
$B=A^{-1}$
Work Step by Step
$A=\begin{pmatrix}1&-1\\-1&2\end{pmatrix}\quad B=\begin{pmatrix}2&1\\1&1\end{pmatrix}$
To verify that $B$ is the inverse of $A$ we simply need to perform matrix multiplication:
$AB=\begin{pmatrix}1&-1\\-1&2\end{pmatrix}\begin{pmatrix}2&1\\1&1\end{pmatrix}=\begin{pmatrix}1\times2+(-1)\times1&1\times1+(-1)\times1\\(-1)\times2+2\times1&(-1)\times1+2\times1\end{pmatrix}$
$=\begin{pmatrix}1&0\\0&1\end{pmatrix}$
$BA=\begin{pmatrix}2&1\\1&1\end{pmatrix}\begin{pmatrix}1&-1\\-1&2\end{pmatrix}=\begin{pmatrix}2\times1+1\times(-1)&2\times(-1)+1\times2\\1\times1+1\times(-1)&1\times(-1)+1\times2\end{pmatrix}$
$=\begin{pmatrix}1&0\\0&1\end{pmatrix}$
and so $B=A^{-1}$
