Let A be a matrix, and let x and y be linearly independent vectors such that

Let A be a matrix, and let x and y be linearly independent vectors such that
\(Ax=y,\ Ay=x+2y\)
Then we have that
\(A^5x=ax+by\)  for some scalars of a and b. Find the ordered pair (a,b).

\(\begin{array}{|rcll|} \hline \mathbf{Ax} &=&y \quad & | \quad \cdot A \\ A^2x &=& Ay \quad & | \quad \mathbf{Ay =x+2y} \\ &=& x+2y \quad & | \quad \cdot A \\\\ A^3x &=& Ax+2 Ay \\ &=& y+2(x+2y) \\ &=& 2x+5y \quad & | \quad \cdot A \\\\ A^4x &=& 2Ax+5 Ay \\ &=& 2y+5(x+2y) \\ &=& 5x+12y \quad & | \quad \cdot A \\\\ A^5x &=& 5Ax+12 Ay \\ &=& 5y+12(x+2y) \\ &=& \mathbf{12x+29y} \\ \hline \end{array}\)