<s>[INST] 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.
 [/INST] Hint: we have
$$A\begin{pmatrix} 7\\-1\end{pmatrix}=\begin{pmatrix} 6\\2\end{pmatrix}$$
And therefore
$$\begin{align}A\begin{pmatrix} 6\\2\end{pmatrix}
&=A^2\begin{pmatrix} 7\\-1\end{pmatrix}\\
&=A\begin{pmatrix} 7\\-1\end{pmatrix}\\
&=\begin{pmatrix} 6\\2\end{pmatrix}
\end{align}$$
So if $A=\begin{pmatrix} a&b\\c&d\end{pmatrix}$ we have
$$\begin{cases} 7a-b=6\\7c-d=2\\6a+2b=6\\6c+2d=2\end{cases}$$
And this equivalent to two systems of two equations with two unknowns $(a,b)$ on one hand and $(c,d)$ on the other and I am sure you can solve.
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