5

B fc) AI j- fo) -A 4 fka) U~ 1>J+ fo)= R 4 4 (a)(() Ixi , *fo HLw; for Fri 0 , ^ = 0 Ikh...

Question

B fc) AI j- fo) -A 4 fka) U~ 1>J+ fo)= R 4 4 (a)(() Ixi , *fo HLw; for Fri 0 , ^ = 0 Ikh

B fc) A I j- fo) -A 4 fka) U~ 1>J+ fo)= R 4 4 (a) (() Ixi , *fo HLw; for Fri 0 , ^ = 0 Ikh



Answers

If $\mathrm{A}=\left[\begin{array}{ll}0 & \mathrm{i} \\ \mathrm{i} & 0\end{array}\right]$, then $\mathrm{A}^{4}$ is (a) I (b) -I (c) 2I (d) $4 \mathrm{~A}$

So this problem we have the following matrix he's equal to for negative 100 0000 00 to negative three. And finally, for the last rose, you go 01 negative too. And we can obtain the care Chrissy equation in terms of Lambda so we can get that using the properties of the determinant of a minus Lambda Chi And we can obtain this as once we do the competition's we will get four minus lambda times Negative Lambda Times two minus lambda times Negative to minus Lambda plus three is equal to zero. So from this factored expression or partially factored expression, we can get Lambda one X equal to four Lambda to equal zero limit of three equals negative one and Lambda four equals one so we can compute the argon vectors as we've been doing in the previous problems. As again as you one is equal to P. Times 1000 where here pees just ah non zero real number, much like R s in the previous problem. And you too is equal to T times 1400 You three is equal to s Let's just say s the letter after? Doesn't really matter. It's just a non zero real number. It's like a placeholder here, and you four finally is equal to your four is gonna be equal to s times 001 and 1/3. So we can have our fundamental matrix composed from these wagon values and Duggan vectors. And we have to accept t The fundamental matrix is gonna be given by you to the 40 100 0400 00 each, the negative T t and 00 each that negative t 1/3 you need to the t.

We want to get the fundamental mattress off the system. Exodus shifty equals the method X a multiplied by X 50 and we are allowed to use a computer software to calculate the Egan varies on the Yangon victors of the metrics. By using a software we can get that the island values of the metrics or equals post minus one. These are real Eigen values, and we have complex values where r equals who plus three i two plus or minus three I and these are the complex. I am values we can get from this complice Eikenberry that all four equals two real bird and B equals three, which is a compression of the imaginable. And using a software we can get the again. Victor's corresponding to each Eigen value at r equals one. They have Eigen Vector you equals 1111 one on one and at R equals minus one, we have the again Victor one minus one, one minus one. And for the complex again, Victor Eigen values R equals two less Really. For example, we can get that equals one two plus three I minus five plus 12 oy minus 46. Last night and I and we can spirit the report and the complex port. The report is one Tu minus five, minus 46. And the measure Bert 03 12. And my and we call this Victor A. And this Voeckler is called the That We can move on to get therefore solutions of the system. Where except one of t equals see one. But the boy boy eat is about are we start by the real constant. We start by the Rinne Eigen values. We have the first Eikenberry one. It was about a rt. We have our one Then It's either a lot of tea but the blood by the victor you which was 1111 The second solution equals C two e. There's a lot of rt we have now are equals minus one. This is minus t live by the victor You To which waas one minus one one minus one. No, we get Take a three off the using oil This formula C 31 tabloid boy, cause I in first we have it was about of all Fatih as we remember that we have alpha equals two is alpha and beta a called history we have It was about all 40 duty, but the blood by cruise line May 30. We have better three while the boy was a victor A that we have e one to minus five minus 46 plus no. These mines minus. He was about of all 30 science May 30 the blood by the victor being now we have the 03 12 9 and similarly we can get next four equals C four multiplied boy the same as its three Andriy Switch designs on science and switch The sign in between We have either or to t sign three t to blow by the victory one to minus five minus 46. Plus, he was about off duty cause I in pretty buoyed by the victor B 03 12 line. Now we can get the fundamental metrics. We have four victors. The first victor's comes the first victor come from X one the second x two and so on. Then we have the fundamental metrics x off the equals. It was about of tea. Either a lot of the either or tea. It's raw. 50 from X one from X to eat is about off minus T minus. It is about a dynasty, each of our finest e minus eaters or by S T. And the third victor is it is about off duty design three D, then minus five. It is about of two. T Resign three D minus. It is a lot of two. D minus three is about to tee Sign three t Andi, No, look, look minus that. Said who He's not minus five is to. And the third a number. The third element is minus five. It was about of two t cause I three t minus 12 eaters about turkey sign three D finally, minus 46. Eats about off duty cause I three D minus mine. Neither were off duty Sign Pretty. And the fourth victor is it is a lot of duty Sign three d. We can switch the science and resigns And the sign in between for this vector for easier Caucasians, Then we have to. It was a lot of duty saying 70 plus three years or 30 cause I in three D and we have minus 5 ft to the world of two. T sign, sweetie Plus will. It was a lot of duty cosigned three t and finally we have minus 46 a drawer off duty, cause I in three d bliss nine. It's a lot of two t cause I in three D, and this is a final answer off our program.

Right. So you tried to sell beans using you have just two. So yeah, I have That's a minus. All right. One where I why you have All right. Uh, what I want So going in What? Just under, well, decide if you want. All right. Uh, what? No, The only difference here, this is a well or what's minus. All right. You that, Uh what? What's with 44 by two for my what's 44 four minus 44. You know what? Very similar answers, but you'll notice switch. So what has no value? I

And this a question. Simplified each of suppression, extra precious teams alike. But the position off the exponents negative toe difference for in each part. So looking for first at parts A and would they have for months go by quickly by white under the negative export? And it's it's a clear that did, Nichols was explained, is only for why. So why is the based off? This explained now? I don't want the negative side to exist in the solution, so I have to take the reciprocal off. Why alone? Because why is alone based off this power? So four X will remain at the top. Why now is indeed a monitor on D minus two will become positive toe. So this is a solution forthe day. Let's move to part B in part B. The negative, too, is for the whole expression for four x four ex wife. Ah, so you can notice that How much did how d stood are different? Because the four ex wife is now inside two brackets. So this is Didi's our debates off this power. So I want to take this broken off the whole expression and other to get rid off the negative sign here. So it becomes one over the hole suppression for eggs y and the hole is to power off. Two. So now you can also ah, distribute this negative. This ah, power off to the exploding toe inside the brackets so it would become a multiplied by each explained inside the brackets with are formed by or fund export Want white dwarf one so it will multiply by each one here. So this is this is one over four to the power off, one multiplied by two is four part off too and we know that it's 16. So four months of power off one multiplied by two and X will be one multiplied by two as well will be too. And why also it won over one multiplied by two, which is too. So I distributed the external explosions inside the brackets so it will be brought about by each internal explained of this expression. So this is the solution. Now let's move in two parts. See where there is the four export off minus two by y and the minus two is now for all the ex. So I want toe get the reciprocal off ex alone on This is four. Why will remain and the numerator and ex become Indian. Memory denominator, Andi explained. Would become positive for party D minus two is only for Dick Concept number four. So I want to take the reciprocal off. Four. So x Why will remain at the numerator and four is now 44 2 and forth Board off. Do we know that it's for much, but by itself twice. And this is 16 so these are the solutions.


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