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Present at least two methods for computing all arbitrary power of the following HOn-dliagonalizable matrix 12 25 and show that the results given by different metho...

Question

Present at least two methods for computing all arbitrary power of the following HOn-dliagonalizable matrix 12 25 and show that the results given by different methods are the same. Please exclude the naive method (multiplying the matrix many times without using its characteristic equation) . Discuss pros and cOnS The following formulas can be used without proof:mA"= (4 ;)" = (& Am ') 2A +1)A" mA"-1 0)" = ( ~mA"tI ~(m -IA" (9 2 )" = (-nx -4)A"

Present at least two methods for computing all arbitrary power of the following HOn-dliagonalizable matrix 12 25 and show that the results given by different methods are the same. Please exclude the naive method (multiplying the matrix many times without using its characteristic equation) . Discuss pros and cOnS The following formulas can be used without proof: mA"= (4 ;)" = (& Am ') 2A +1)A" mA"-1 0)" = ( ~mA"tI ~(m -IA" (9 2 )" = (-nx -4)A" ~mA"+! mA"-1 (m + 1)A"



Answers

The matrix $M=\left[\begin{array}{cc}2 & 1 \\ -3 & -2\end{array}\right]$ has some very interesting properties. Compute the powers $M^{2}$ $M^{3}, M^{4},$ and $M^{5},$ then discuss what you find. Try to find/create another $2 \times 2$ matrix that has similar properties.

Hello is discussion. We have a scram interest. A Onda. We have been told to square this metrics than find a cube of this metrics. And in the fourth power and so on and ultimately arrived to adjourn from love for 84. And so let us for square it and then see what we get. So is square close. 1101 110 So we know that we have to first multiply the first true and the first column toe Get the first element off the resulting metrics, So multiply and then edit. Okay, so we will get one month. A good one. This one then one market big crazy Who is zero then again 20 Playing the first true In the second column we would get the second element off the resulting matches soon. Money plays one with one Give us one. And then again one, it's one. Similarly, we will get here zero zero, then zero Yes or file Metrics Wickham's one do see you. Okay, so this is now we will find a que It is a square into it. So one do 01 one one single. Similarly, if we multiply each element. And then Ed, as earlier we'll get firstly went would be one plus geo saying element would be one less too. Parliament would be zero plus zero and the 4th 1 week Cedo less one. So we will get one Zito So 13 Okay, so we can see he was 1101 This is so into the one Then E square, we found one to cedar one And then now CQ, we have calculated 13 CDO. If you notice the three matrices, click carefully. You can see this part is common to all. This is common. The only changing thing is this one. And you can easily map this part with this hour, isn't it? So we can come up with the general equation off e to the power em is 101 The common part and this world would be. And so then we have it. Thank you.

So for this problem, we're going to be calculating each of the powers of A until we are able to determine some some pattern for our product. I'm so we know the age of the first power. Is this to us, you matrix? That's entirely made up of ones. So if we want to find a squared, we're just going to multiply this matrix by itself. I'm so the matrix 1111 by the Matrix 1111 And we know that in order to calculate this, we're just going to compute the inner product of several times. So we'll start with the first row of eight in the first column of our second, eh? We get one times one plus one times one I'm so one plus one is just going to give us too. I want that next. We could do the same thing for our second column of our second Matrix. And again we have one times one plus one times one which will give us, too. And then finally, for our third row or a second row of our first Matrix and her first column of our second matrix, we again get one plus one which is to and this will be the same thing for our second column in second row. That's another two. I'm So now we have a squared. We could go ahead and calculate a cube, a cube. We know we're going to use our two by two matrix a squared, and we'll try this by another. A matrix, which was 1111 And so in the most by this together, um, for our first row of a oh, our first or of a squared in her first column of a we get two times one plus two times one which is going to be four. And then this will be the same thing for our second column of our second matrix two plus two well enough before again and then for our second row of a of a squared times. Our first column of a begin get four. And this is the same thing for our second column so we can see that we are essentially going to be squaring our terms here. But I'm going to go ahead and calculate another way to the fourth Pepper. I'm so this is going to be a cube which we know is a square matrix made up of force. Time's a 1111 Um, and once again, in our first row of a cubed times our first column of a we're going to get four plus four. So that will be eat now then, since we can see that we're all flying by the same a number in each of our columns of a any tour Rosa A. That tells us that each entry here is going to be eight. I'm so something interesting we noticed, is that we are going to increase at it an uneven amount. So the increase from a to the first power to a squared is going to be a one number increase. But a square to a cube is a to increase. I'm so one interesting prior that we noticed is that these are all going to be powers of two. I'm so our A to the first power matrix is made up of two's TV zero power. A Mexican we know anything to zero power is one a squared weaken, right as to to the first power a cube. We know that two squared is four. So this will be to the second power. And finally age of the fourth power. We know that eight is actually equal to two cubed s O to cube, to cube, to give and to cubed. So, essentially, our general formula that weaken right here, um, is for some excellent it? Aye, to the end power. We know that this was first power, second power, third power and fourth power. So a to the N power is going to give us a two by two matrix made up of two raised to the N minus one power in each of our entries in this matrix.

Everyone. So in this problem, we will be performing this mattress multiplication off a with itself and obtained the different powers off a and then see whether there is a pattern and lusty will arrive to a general formula off the poor. Okay, so this is a pretty poor and now is a square matrix whose all the elements are one. Okay, so, no, let us find a square. The square is a into it. That would be equal to 1111 We should be market black with again 111 So the rule for multiplication is dead. We will take the first row off the first batteries and then multiply its elements with the elements off the first column of the second mattress and then add whatever multiplied value is up there. Okay, so by that, I mean, one would be multiplied with this one. Then this value will be added to the motive, like well off this one. But this one Okay, this will form the first element off our has certain two by two matrix. Okay. The resulting metrics will also be a who by too much is okay. So that would be one plus one. Similarly, if you see all the elements in the resulting metrics will be one less one. So that is to do do. And so this is is under this Fine. Thank you. So a cube is a square multiplied by. Okay, So what is this square do who? And he's 1111 So similar rules should be applied for multiplication. And we your opt in two multiplied by one is too. Then again, to multiplied by one is too in a similar way. All other elements will be do. Plus So this would give us four. I won't up with this spired we could not get figure out what is going So let us begin perform and find into the golf four, which is Thank you. So a cube is four more more and food and he's 111 So this is four again. Four multiplied with one. And this forest manipulated this one. And if added, they will give us eight Similarly all other elements. We also be okay. Okay, So this is our eight to the buffalo. So if I write all the values Upton over here so that it to the bone one was 1111 It is important to Waas who? Who? I'm to Italy. Both three was a metrics with all the elements food and Leslie into the bowl full had all elements. Eight. Okay, so if you see the elements are same in all the mattresses. Okay. The individual practices have all the four live unseen. So if you see the first metrics, this is Italy one and elements are the 2nd 1 is a square the limits of or do the card. One is a cube. Elements are all four and the last one is Italy powerful. The elements are all eight. If you see the elements, you can see that there any better off the ball off? Yes. These air, in a pattern of power off to the power zero will give you one. So every limitless one who believe our one will give you two. Those two will give you four. And this par three will give you eight in this way that the mattresses are happening. Okay, so how to find it to the poor? So you can see the relation between this power and this power? There's a difference of one. Okay, there is a difference of one. There is also a difference off one. Okay, So, generally into the bar, M would be due to the power and minus one political and minus one n minus one and then again to to the border in minutes. So this is the generally question for it. To the ball. Thank you.


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