How do I find this vector?

Let \(A\) be a matrix such that \(\mathbf{A}^{-1} = \begin{pmatrix} 1 & 1 \\ 2 & x \end{pmatrix}\)
for some value of \(x\) .
What is the vector that  maps to \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\)?

I assume:

\(\begin{array}{|rcll|} \hline Av &=& \dbinom10 \quad & | \quad \times A^{-1}\\ A^{-1}Av &=& A^{-1}\dbinom10 \quad & | \quad A^{-1}A = I \\ Iv &=& A^{-1}\dbinom10 \quad & | \quad Iv = v \\ v &=& A^{-1}\dbinom10 \\\\ v &=& \begin{pmatrix} 1 & 1 \\ 2 & x \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix} \\\\ v &=& \begin{pmatrix} 1*1+1*0 \\ 2*1+x*0 \end{pmatrix} \\\\ \mathbf{v} &=& \mathbf{ \dbinom12 } \\ \hline \end{array}\)