<s>[INST] Find the matrix $\mathbf{A}$ if $A\binom{7}{-1} = \binom{6}{2}.$ Find the $2\times2$ matrix $A$ where $A^2=A$ and
$$A\begin{pmatrix} 7 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}.$$
I tried plugging in: $A= \begin{pmatrix}a&b\\c&d\end{pmatrix}$ but that became messy very quickly. I got the equations:
$7a-b = 6$
$7c-d = 2$
$a^2+bc = a$
$ab+bd = b$
$ac + cd = c$
$bc + d^2 = d$ from trying that method. What should I do? 
 [/INST] As $A^2=A$, you also know that $A\begin{pmatrix}6\\2\end{pmatrix}=\begin{pmatrix}6\\2\end{pmatrix}$. Since $\begin{pmatrix}7\\-1\end{pmatrix},\begin{pmatrix}6\\2\end{pmatrix}$ are linearly independant, this determines $A$ completely. We conclude that both columns of $A$ are multiples of $\begin{pmatrix}6\\2\end{pmatrix}$, so
$$ A=\begin{pmatrix}6x&6y\\2x&2y\end{pmatrix}$$
for certain $x,y$. Now $A\begin{pmatrix}6\\2\end{pmatrix}=\begin{pmatrix}6\\2\end{pmatrix}$ just means that $6x+2y=1$, and $A\begin{pmatrix}7\\-1\end{pmatrix}=\begin{pmatrix}6\\2\end{pmatrix}$ means that $7x-y=1$. Can you find $x,y$ now?
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