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$\mathbf{A}=\left[ \begin{array}{cccc}{0} & {1} & {0} & {0} \\ {-1} & {0} & {0} & {0} \\ {0} & {0} & {0} & {1} \\ {0} & {0} & {-1} & {0}\end{array}\right]$

So in this problem we're given this matrix which Is a diagonal, right, only has entries on the main diagonal. Everything else is zero and were asked to define the determinant. We can use a matrix calculator to do this with. So what you desmond dot com went to math tools matrix calculator and got this one. So I need a new matrix now and I got five rows and five columns. And the first entry up here Is a -2 And then the entry here is a three And the entry here is a -1 man. She just working my way down the main diagonal here, This is a two And the last entry down here is a -4. There's all my entries now to do the determinant. I go d E t of a and We get -48 for the answer. So there you go. Which by the way, if you multiply the entries on the diagonal there. Look what happens When one is 2 times three is -6. I was in -1 is plus six times two is 12 Times of -4 is -48. Gave us the determinant, didn't it?

Okay, So this problem again, we're getting this big five by five meet tricks and are told to you some software to finer Eigen values and vectors. So the Eigen values that I got were one negative one negative one. I am negative by in the I again. Back days. Now that I got this one 01000 00100 a 0001 I and then a 0001 and negative. I okay. And then the big X t that I got from the software is is harder t followed by four zeros than zero one plus t. Either the negative t a negative t he wasn't a t Syria zero zero. He eat negativity. One minus t e to the negative. T 00 There is your is your, uh, co sanity. Negative sign of tea. Use an A T. No sign of T and three more zeros. Okay, so when we violate this at zero, all of our science will turn zero. Everything multiplied by t will turn to zero and we get when I get the identity matrix okay? And says nice thing about that. You could just fill in the rest with zeros There is that that means are inverse is also the identity matrix. So when we want to quickly by hand, that eat out of a T being equal to, you know, x of t times that in verse at zero. Ah, the inverse being the attorney matrix just means that the answer is our except E. So we don't have to do any calculating. We already have that from our software, and that's gonna be the answer.

Okay, So he were given this matrix five by five a and are told to find its Eigen values using some sort of software like Matt Lab. And when we do that, we will get Eigen values of negative 13 times and then negative, too. Ah, Twice. Okay. And then we will also use this software to find the I again vectors her the exercise prompt and nose. No more space to write him. Ah, those air going to be one with for zeros. Um, then we have another one with what is in these second row, we'll have one in the third row, a one in the fourth row. And then to no one's surprise, we get a one in the final. They throw okay and says it will also use our software to you find the x of t and we do that get ah, big kind of gross matrix. But it's going to be one plus t because 1/2 a t squared times either negative t then they will have Ah, negative 1/2 a t square times he the negative t on a negative t plus 1/2 a t squared, uh, times eating it. You know why something changed my way of writing? 1/2 But just go with it. Um, And then the next room, we're gonna have the next column t plus t square times e to the negative t one plus t minus t squared times each the negative t on negative three t first he squared times either the negative t and a game and that with two zeros. So our third column will be 1/2 t square times e to the negative t t minus 1/2 t squared times e to the negative t and then one minus two t plus 1/2 t squared times either native tea and two zeros. And so, no, I've run out of room. So just imagine, um, that the next to rose air kind of continuing below here in the next two columns. And so for those will get zero is a top and then ah, to t plus one times e to the negative to t negative for tea times. Either native T T. And our final column lives years atop a swell and then tee times e the negative to t and a negative to t plus one times e to the negative to t No. We put zeros in here to get our x zero all the tease about zero of the heat of the power of two people, but one. And we actually end up with the identity matrix one to calculate all that out. Right, So we would have one appeared get one plus zero plus zero is one times one. Um, but the next we would give course 00 there. Same with the 3rd 1 Wind up with zero. So when we put you in for tea and calculate I didn T Matrix is what we get If you got something else, just recheck your calculations, okay? And so that means that if we've got any matrix, then of course, the inverse is also gonna be the identity matrix, right? So when we go to solve that E to the power of a T is equal to x t, uh, times the inverse of that zero. Since the inverse is at zero, it's just going to be this they expertise. We don't have to do any of the multiplication of that step. Ah, we have the answer here from before, and that will be our final solution.

All right. So, again, we're tasked with finding, um so alien values and vectors using some sort of software, their choice, Something like the egg function. And matt Lab works well, um, and so this is our given matrix A here. And when we plugged that in two, some software and we look for the again values, we end up with negative one three times and negative to twice. Okay. And then we also get for our again vectors. It's quite similar to the previous problem, but so look, a a one in the first row, one in the second row, a one in the third row, I would buy just a one in the fourth row and finally just the one in the final Three throat. So there are in vectors and I in values. And then we also use this software to get I big X of t main tricks. Okay. And so what we get fat is will get you to the negative t and then but zeros, um, zero and then a one plus t. Either the negative t and negative tee times e to the negative t 00 and then we get zero tee times e to the native t a one minus t times e to the negative t 00 three zeros followed by a to T plus one e to the negative to t and negative for tea times e the negative to t And lastly, another 30 is a top tee times e to the negative to t and then a negative to t costs one times e to the negative t sort of another room there. But you need it, okay. And so that now we do the fun part of evaluating it. At T equals zero. So t equals you will get a one here and zeros town the rest when t equals zero. We end up with a one here in the second column. Second really spot and zeros everywhere else for that column we do. The third column will get a one here in the third row, third column, but zero everywhere else and segment maybe could see where this is going. I will also get ones at the fourth row, fourth column spot and fifth row, fifth columns. That means that evaluated at zero we get, he had under T matrix. Okay, if this is the attorney matrix and the, uh, inverse of it is also the identity matrix. Absolute serious illness, calculating labor, and, ah, that means that we wouldn't go to fine. Are either the heart of 80 being equal to X of t times the inverse at zero. Since that's the identity matrix we get that is just equal to that. The exit T. Right. So the answer is this function that we already got from our software, Not this matrix. Right? Um, here. Yes. So that's gonna be our final answer.


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