Find the 2x2 matrix given 2 equations. Find the $2 \times 2$ matrix ${A}$ such that ${A}^2 = {A}$ and
${A} \begin{pmatrix} 7 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}.$
I have tried to express A as a matrix with variables a,b,c, and d, but it gets too messy with the equations.   Any help is appreciated.

Notice that $A \begin{pmatrix} 7 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}$ and also $A \begin{pmatrix} 6 \\ 2 \end{pmatrix} = A^2 \begin{pmatrix} 7 \\ -1 \end{pmatrix} = A \begin{pmatrix} 7 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}$.
Thus $A \begin{pmatrix} 7 &6  \\ -1 & 2 \end{pmatrix} = \begin{pmatrix} 6& 6 \\ 2& 2 \end{pmatrix}$.
And therefore $A = \begin{pmatrix} 6& 6 \\ 2& 2 \end{pmatrix}\begin{pmatrix} 7 &6  \\ -1 & 2 \end{pmatrix}^{-1} = \frac{1}{10}\begin{pmatrix} 9& 3 \\ 3& 1 \end{pmatrix}$.
