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

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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).

Jul 7, 2019

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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).

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

Jul 8, 2019