$A = \left[ {\begin{array}{*{20}{c}}
  { - 1} \\ 
  4 
\end{array}\,\,\,\,\begin{array}{*{20}{c}}
  2 \\ 
  { - 3} 
\end{array}} \right] \Rightarrow {A^{ - 1}} = \frac{1}{{ - 5}}\left[ {\begin{array}{*{20}{c}}
  { - 3} \\ 
  { - 4} 
\end{array}\,\,\,\,\begin{array}{*{20}{c}}
  { - 2} \\ 
  { - 1} 
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  {\frac{3}{5}} \\ 
  {\frac{4}{5}} 
\end{array}\,\,\,\,\begin{array}{*{20}{c}}
  {\frac{2}{5}} \\ 
  {\frac{1}{5}} 
\end{array}} \right]$

$\alpha A + \beta I = {A^{ - 1}} \Rightarrow \left[ {\begin{array}{*{20}{c}}
  { - \alpha } \\ 
  {4\alpha } 
\end{array}\,\,\,\,\begin{array}{*{20}{c}}
  {2\alpha } \\ 
  { - 3\alpha } 
\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
  \beta  \\ 
  0 
\end{array}\,\,\,\,\begin{array}{*{20}{c}}
  0 \\ 
  \beta  
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  {\frac{3}{5}} \\ 
  {\frac{4}{5}} 
\end{array}\,\,\,\,\begin{array}{*{20}{c}}
  {\frac{2}{5}} \\ 
  {\frac{1}{5}} 
\end{array}} \right]$

$ \Rightarrow \left[ {\begin{array}{*{20}{c}}
  { - \alpha  + \beta } \\ 
  {4\alpha } 
\end{array}\,\,\,\,\begin{array}{*{20}{c}}
  {2\alpha } \\ 
  { - 3\alpha  + \beta } 
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  {\frac{3}{5}} \\ 
  {\frac{4}{5}} 
\end{array}\,\,\,\,\begin{array}{*{20}{c}}
  {\frac{2}{5}} \\ 
  {\frac{1}{5}} 
\end{array}} \right]$

$ \Rightarrow \left\{ \begin{gathered}
  2\alpha  = \frac{2}{5} \Rightarrow \alpha  = \frac{1}{5} \hfill \\
   - \alpha  + \beta  = \frac{3}{5} \Rightarrow  - \frac{1}{5} + \beta  = \frac{3}{5} \Rightarrow \beta  = \frac{4}{5} \hfill \\ 
\end{gathered}  \right\} \Rightarrow \frac{\beta }{\alpha } = 4$