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.

First, $A^2-A=0$ means $A$ must have eigenvalues 1 or 0. $A$ is not a zero matrix, so the two eigenvalues of $A$ cannot be all 0. The two eigenvalues cannot be all 1 either for otherwise $A$ would either be (1) the identity matrix which is certainly the case (2) a nondiagonalizable matrix but judging from the polynomial $A$ satisfies this cannot be the case. This leaves the only possibility that $A$ has eigenvalues 0 and 1, and so it's of rank 1.
Since $A\begin{pmatrix}7\\-1\end{pmatrix}=\begin{pmatrix}6\\2\end{pmatrix}$ its range is $\text{Span\{(3, 1)\}}$ so $A=\begin{pmatrix}3a&3b\\a&b\end{pmatrix}$ for some $a$ and $b$. Its null space is spanned by $\begin{pmatrix}7\\-1\end{pmatrix}-\begin{pmatrix}6\\2\end{pmatrix}=\begin{pmatrix}1\\-3\end{pmatrix}$. So
\begin{eqnarray}\begin{pmatrix}3a&3b\\a&b\end{pmatrix}\begin{pmatrix}1\\-3\end{pmatrix}=0\end{eqnarray}
implies $a=3b$. Finally
\begin{eqnarray}\begin{pmatrix}9b&3b\\3b&b\end{pmatrix}\begin{pmatrix}7\\-1\end{pmatrix}=\begin{pmatrix}6\\2\end{pmatrix}\end{eqnarray}
implies that $b=\frac{1}{10}$. So $A=\frac{1}{10}\begin{pmatrix}9&3\\3&1\end{pmatrix}$
