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Use the factorization $A=P D P^{-1}$ to compute $A^{k},$ where $k$ represents an arbitrary positive integer.$\left[\begin{array}{cc}{a} & {0} \\ {3(a-b)} & {b}\end{array}\right]=\left[\begin{array}{ll}{1} & {0} \\ {3} & {1}\end{array}\right]\left[\begin{array}{ll}{a} & {0} \\ {0} & {b}\end{array}\right]\left[\begin{array}{rr}{1} & {0} \\ {-3} & {1}\end{array}\right]$

$A^{k}=P D^{k} P^{-1}=\left[\begin{array}{ll}{1} & {0} \\ {3} & {1}\end{array}\right]\left[\begin{array}{cc}{a^{k}} & {0} \\ {0} & {b^{k}}\end{array}\right]\left[\begin{array}{cc}{1} & {0} \\ {-3} & {1}\end{array}\right]=\left[\begin{array}{cc}{a^{k}} & {0} \\ {3 a^{k}-3 b^{k}} & {b^{k}}\end{array}\right]$

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 3

Diagonalization

Vectors

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Okay, so we have once again, uh, in a dining dia generalization of the matrix PDP minus one. What we want to know is a general formula for a to the power case of what is a in the power k for Okay, uh, positive integer. So we know that a que is p times d to the power k times p minus Juan based on page 284 of the boat. So we'll just now we plug in the values of P and G, which are given a question. So he is 1031 de is a 00 B. So a diagonal matrix, two of their power K and P minus one is 10 minus 31 So PND are given now because he is a diagonal matrix. What can we do? We can simply ah say that d to the park is a k agent of power K 00 b to the power K because he is a diagonal matrix, P and P minus one don't change. So we just justify what we do because the is ah, diagonal matrix. Perfect. Now what do we do? We just compute this matrix product. So we start with the two matrices on the left. So here we will have a do that power K zero three A to the power K B to the power K. So we just doing normal standard two by two matrix multiplication. And we have P minus one. So 10 minus 31 and we do one more matrix multiplication. So we'll have a que zero three A k minus three b to the power. Okay. And I will just be raised zero and make sure everything is perfectly allying and we'll have B k right here. So the and that's the final answer. So for Okay, positive integer We have a K agent of power K. It is a small age of the power K 03 small age of the power. K minus three. Small beating the power K. Yeah, not zero b. K. And there is your final answer. So what it we used the main theory in to use in that question is the fact that a did it power. Okay, is p times D to the power K times p minus one for any diagon ization of the Matrix

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