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0 i Ta rollewin; Naz(S04)unaed CNnmlm Tamicai Cu(cioshCjinbiclTennicomingMah tha uia hidanen finl 784, Fula Ute nameyour pladuc[h[ylo EorndHalmate IHan Omon_Alaen' & 'Uuodi aibgn]

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$$\left|\begin{array}{ccccr} -2 & 0 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & -4 \end{array}\right|$$

I have a number 43 and making yourself the questions using grand module equations are X plus. Why? Blessed said plus w equal to zero two x place are not blessed. Here it is. Plus w equals zero off. Why miners ed equal to zero and x place does ed equal to one. So we were defining and metrics non is coefficient metrics d as coefficient of X when toe. Okay, first of all, we should write year zero into X. Is he going to? Why zero into is that does he go into the blue zero into? Why? Unless you're going to w no. Right one does it over. One ones little ones. It'll 10 minus 12 1100 Let the Scots ticket 1201 Okay, Great. 1010 Very good. 10 minus 12 Okay, 110 Now let's find this determinant. Fun tau 01 When Zito one zero. When? Zero minus one toe. 110 in zero. So there are two zeros. They're two zeroes in rule number two or number three or number four. So there is the possibility of getting 30 But let's just start by it'll three expending with three. So this is bless. Minus bless minus. So So for the 30 1st will be zero second will be one and the elements will be on The elements will become off one toe, one when toe one 100100 then no, there would be no one's resident because we have to, you know, color number two. So this will simply become once you went to and one one and Zito, they're living. This are three end. This will become 1 to 1 10 to 11 ended. Okay, now comes minus one. Bless miners. Bless. So it should have been negative. Off one No, minus one for this. So we will be having just It's gone. This is gone. 12 one, one Tzeitel, Zito And the one when And Zito and another will be zero. Okay, so this is minus one And we have to know expend, we should expand using column three What? Okay, Colin Civilly Perfect. That is a start expending column C So if you take one, let us star sign convincing plus minus plus minus And plus, if you take this one. The remaining elements will be 2102 Is it We were writing. One, 21 zero and two. If second will be one, then too many elements will be 1112 minus one. So would be 111 and two. Dad will be zero. And for this it will be minus when we should just expend with Colin. Number to replace. Minus place and minus. So column number two means this will be minus one. And the remaining elements will be 2110 to 11 end zero, that's all. Okay, so this will become minus one and going to do four minus zero, minus one. Tu minus one. Is it blush zero minus one that is minus one. This will become minus one into four, minus one. Eso for minus one, minus one. That is my three miles one, which is in minus four. So we have determinant. Be equal to minus four. Okay. Malcolm's DX, that is determinant. A Matics, often by the placing first row. Also I first column with and I didn't say Don't give any questions. That 0001 2nd and third will be the same. That is 1010 2nd, 3rd and 4th William in the same 1012 This is 110 n zero. They just find out its determinant. 00011010 1012 and 110 in zero. Well, it is a started with column one expendable caller, one plus minus, plus minus. That will be minus one. And the remaining elements will be 101 one's a little one and 11 Zito. Okay, there is no mistaking. No altering. The sign minus one. OK, they should be in minus one into a 10 minus one in tow. 1100 K minus one and two. So this will become 10 minus 1101110 Now expending using that word to bless. Minus place minus. This will be minus one for the 00 and minus. When so very many elements will be one minus one. No one and one when this is one plus one, that is to so determinant off X will be equal to the dominant x will be called to now comes become minute off. Why O Matics doing metrics? So this will be often by replacing Second Column using 0001 That's the first condominium in the same one toe zero and 10 Okay, we're replacing. So 00 Is it a 10001 thirties, 10 minus one toe. Four days. 11 zero n zero. So it's a big determinant. The way one toe 01 0001 10 minus 1 to 11 zero Zito since the three zeros here. So let us start with this expanding using column number two. So they said, Bless. Minus, plus minus and plus. So for this it will be zero for this also. Zero for this is also zero. So for the one vision left with the elements, this will be 1201 toe zero 10 minus 111 zero. Okay, so now Dera two zeros in row number three. So let's expand using or number three plus minus, plus minus And plus So one for this. It will become zero for this. No, it's OK. Ok, Took it one more time. Okay. This is minus one into miners and the remaining elements will be this. This is gone. This is gone. So one minus two, one minus two. And the one and one No. Minus misplace one. So one into one one plus two. That is three I met, one less toe. Three. So there is minus minus. Plus one here. One thing to be noted that I have made. This should be blessed to on Lane, not minus two. Because this sign, it's only was defend in the extension is using this column, Audra or including this element. So this will be simply one minus door. Oh, Okay. Okay. So this is May one minus two. That is minus one. So do you are equal toe minus one. Now comes days it These are metrics will be opened by first. We should find out the metrics. Then we will move toward the determinant. So, are these edible? Be open by replacing Third column Tired column by 0001 1st and 2nd William And as it 1st, 2nd, 4th women as it is 120 Fun of 1010 Then 0001 And then 110 and zero. No determinant off days, Ed. Well, Bay 1201 10 ones. It'll video 001 11 and zero and zero that our trees, videos, zeros in here. So we'll be using column three. Bless, minus place minus plus and minus. So everything will become zero. Except when we use this. So remaining elements will be 120120101 11 ends, you know, 11 end zero. Now we will be using Ah! Ah! Okay, so three. Because their two zeros So this would become plus minus plus minus and plus should be one. Do minus one remaining limits. We want to one in one. So this is minus one of one minus two. That is minus one. So it will be one. So we have d said equal to one. One more thing we have to find that is indeed the blue that it last The last column will be replaced was it was either 11 Everything will remain the same. 1201 1010 10 minus 12 and last 3000 and one. So that is define all it is find out the metrics. Mattis, we have already found out the determinant. One toe 01 10 and zero, 10 minus one and two and 000 and one. We will be using column number four. So plus minus, plus minus, plus minus and place. So this will simply become one on remaining elements will become Oneto one, as I want to. Zero 12010110 minus one. Now we will be operating. I spend now expanding using our own number two. So bless minus plus and minus. So minus two 111 and minus one. So this really minus two. Minus one minus one. That is minus two into minus two. Equal to four. No, minus two and minus two will be equal to minus four. What This? Let us try it. Minus two and minus two. Equal Do. For so we have need of looted. The minute. Equal to four. Now let us start X equal toe de exit. Opponent by determinant. So DX determinant days the exit amenities to and is minus four DX is too, and D is minus four. Okay, so this will be Dubai minus for that In minus one by two. Over I will be Do you are determinant by D That is de ways minus one by minus four. So mine is, um, by minus four, minus one by minus four that is won by four. As that is, Dee's a determinant by determinant. So these a determinant ease when by minus four that is minus one by four. And w will be equal toe redoubled, imminent by determine it. That is, deep blue is for by minus four four by minus four. That is minus one. So our answer says X equal to minus one. Lie to why call to one by four as Eric will do minus one by four and a blue Kal toh minus one. These are dances. Thank you so much.

This is our question Number six we had right the augmented mattress for this human system or the question. Now, as you can see for the first column, your coefficient of X y and that that is negative to negative one. Then never do you want when they get you to nobody wants and nobody one on the right side. We have five. No, for the second equation, your coefficient of X y and then that is 11 in one. You're 11 and one on the right side. We have zero now for the integration. Your coefficient off next y and then that is 3 to 7. Your tree. Do any seven when the right side we have negative rate. Well, this is there are of your document makes. Thank you.

Good day, ladies and gentlemen, today we're looking at Problems 25 which have balls solving the initial value problem stated. And, um, this is basically I've broken this into three steps with a bunch of sub steps. And so the first part that we look at is we observed that since this is a the non homogeneous equation induced by this guy, I I this this year, um, can we? The form of it allows us to use the method of undetermined coefficients. And so our first, the first part of solution, is just applying the method of undetermined coefficients, which, of course, is first to solve the home of genius equation. Um, Z double Brian plus C equals zero for the induced homogeneous equation. And when we do that, we first look at the roots, which in this case are pure, complex, plus or minus I. And that tells us the general form of the solution is C one co sign t you proceed to sign t. Okay, so we've now solved a homogeneous equation and for the next part, we look at the Nano Maje being induced. Nono, my genius equation and we find a particular solution So in this case, um, we observe that. Ah, the the non over genius equation. Um, little laugh of tea, which in this case, is this guy here, um, is not a linear combination of y of h, which is this here? Army. So since it's not a linear form or linear combination of why of age, we can go ahead and use. Um, why the particular form here? A czar? Guess so. We can use that as our guests and we do that. And cat, we first calculate a y p prime. And why the double prime? And we input it into here and from. And then we wait. We obtained a linear system of equations which in this case, is just too, eh? Ah, equal that to which then gives us a cool one. So that tells us, then, that the general form of the solution or the general solution of the non homogeneous equation, which is just a linear combination of the home of genius plus, uh, you know, the home a genius here, Uh, plus the particular all right forms a general form of the, uh, not homogeneous. So are the general form of the Yeah, Nono. Virginia solution. Um, And now from there we plugged out in and, uh, look at the the initial value problem. Okay, so, um, we apply the initial conditions, which, in this case, ah Wai of zero. So plugging in. Why of zero into this guy, of course, is gonna give us C one here, zero here and on. That should be one. I'm sorry that that's just too. It should be one. So this should be one, not two lips. Okay. And now I'm making my life more complicated. Um, yeah. So Oops. Ah. OK, so sorry about that. Uh, sign t lost your negative, too. Yeah. Um, so we plug in zero here. We're just going to get C one plus one, which eyes this equation and similarly applying it into the why Prime. And we get C two minus one equals zero. Which, of course, yields the initial value for the solution of initial value problem, um, to be this guy. And, um, the final stuff, which it always should be, is check. That's straightforward. But so that's that's our solution for today. Um, thank you very much.

So in this problem, we're given this matrix and were asked to use the matrix capability of a graphing utility to find the determinant. So I went to Desmond's dot com, went up here to math tools, matrix calculator, and got this matrix calculator here. So in our new matrix and our matrix is a four by four. So four rose and four columns. Now the entries one minus one, C A A and four and two and six zero my S four and then to zero two and six and then zero two, eight zero. Okay, where's my matrix now? Just do D E T right here to get the determinant determinant of a yes, Negative,


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