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Linear Algebra

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Diagonalize to Find High Power

LINALG-XCY3WL

If:

$$A=\begin{pmatrix} 3 & 1 \\\ 0 & 2\end{pmatrix}$$
$$D=\begin{pmatrix} 3 & 0 \\\ 0 & 2\end{pmatrix}$$
$$P=\begin{pmatrix}1 & 1 \\\ 0 & 1\end{pmatrix}$$

...and:

$$P^{-1}=\begin{pmatrix} 1 & -1 \\\ 0 & 1\end{pmatrix}$$

...then $PAP^{-1}=D$.

Given this information, determine $A^{10}$.

A

$\begin{pmatrix} 3^{10}+2^{10} & 0 \\\ 0 & 3^{10}+2^{10}\end{pmatrix}$

B

$\begin{pmatrix} 3^{10} & 3^{10}-2^{10}\\\ 0 & 2^{10}\end{pmatrix}$

C

$\begin{pmatrix} 3^{10} & 2^{10} \\\ 2^{10} & 3^{10}\end{pmatrix}$

D

$\begin{pmatrix} 2^{10} & 0 \\\ 0 & 3^{10}\end{pmatrix}$