Let $\mathbf{A}$ be a matrix, and let $\mathbf{x}$ and $\mathbf{y}$ be linearly independent vectors such that \[\mathbf{A} \mathbf{x} = \mat

Let \(\mathbf{A}\) be a matrix, and let \(\mathbf{x}\) and \(\mathbf{y}\) be linearly independent vectors such that
\(\mathbf{A} \mathbf{x} = \mathbf{y}, \mathbf{A} \mathbf{y} = \mathbf{x} + 2\mathbf{y}\)
Then we have that
\(\mathbf{A}^{-1} \mathbf{x} = a \mathbf{x} + b\mathbf{y}\)
or some scalars a and b.

Find the ordered pair (a,b).

\(\begin{array}{|rcll|} \hline Ax &=& y \quad | \quad \cdot A^{-1} \\ A^{-1}Ax &=& A^{-1}y \quad | \quad A^{-1}A = I \\ x &=& A^{-1}y \\ \mathbf{A^{-1}y} &\mathbf{=}&\mathbf{x} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline Ay &=& x + 2y \quad | \quad \cdot A^{-1} \\ A^{-1}Ay &=& A^{-1}x + 2A^{-1}y \quad | \quad A^{-1}A = I \\ y &=& A^{-1}x + 2A^{-1}y \quad | \quad \mathbf{A^{-1}y=x} \\ y &=& A^{-1}x + 2x \\ \mathbf{A^{-1}x} &\mathbf{=}&\mathbf{-2x+y } \\ \hline \end{array}\)

\(\mathbf{(a,b) = (-2,1)}\)