MATH221 December 2008
• Q1 (a) • Q1 (b) • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q4 (a) • Q4 (b) • Q5 (a) • Q5 (b) • Q6 (a) • Q6 (b) • Q7 (a) • Q7 (b) • Q8 (a) • Q8 (b) • Q9 (a) • Q9 (b) • Q10 (a) • Q10 (b) •
Question 05 (b)

Let $A={\begin{pmatrix}5&0\\2&1\end{pmatrix}}$. Find the matrix $A^{1000}$. (Simplify your answer as much as possible.)

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

Make use of the result of part (a) to diagonalize $A$. Powers of diagonal matrices are easy to compute.

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Solution

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From part (a), we know that the eigenvalues for $A$ are $\lambda _{1}=1$ and $\lambda _{2}=5$, with corresponding eigenvectors $v_{1}={\begin{pmatrix}0\\1\end{pmatrix}}$ and $v_{2}={\begin{pmatrix}2\\1\end{pmatrix}}$, respectively.
Let $D={\begin{pmatrix}1&0\\0&5\end{pmatrix}}$ denote the matrix of eigenvalues, and $P={\begin{pmatrix}0&2\\1&1\end{pmatrix}}$ the matrix whose columns are the corresponding eigenvectors. Then $A$ is diagonalized via $D=P^{1}AP$, or, $A=PDP^{1}$.
This decomposition allows for easy computations of powers of $A$, as for any integer $k$, we have $A^{k}=(PDP^{1})^{k}=PD^{k}P^{1}$, and $D^{k}={\begin{pmatrix}1^{k}&0\\0&5^{k}\end{pmatrix}}$. Therefore, we compute $P^{1}={\begin{pmatrix}{\frac {1}{2}}&1\\{\frac {1}{2}}&0\end{pmatrix}}$, and multiply the three matrices to obtain
$A^{k}=PD^{k}P^{1}={\begin{pmatrix}0&2\\1&1\end{pmatrix}}{\begin{pmatrix}1&0\\0&5^{k}\end{pmatrix}}{\begin{pmatrix}{\frac {1}{2}}&1\\{\frac {1}{2}}&0\end{pmatrix}}={\begin{pmatrix}5^{k}&0\\{\frac {1}{2}}(5^{k}1)&1\end{pmatrix}}.$
Then for $k=1000$, we have $A^{1000}={\begin{pmatrix}5^{1000}&0\\{\frac {1}{2}}(5^{1000}1)&1\end{pmatrix}}.$

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MER QGH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER Tag Eigenvalues and eigenvectors, MER Tag Matrix diagonalization, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag

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