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Q3a) Applying Gaussian elimination wilh and without partial pivoting and digit floating point arithmetic, solve the following system and compare the results_ Show a...

Question

Q3a) Applying Gaussian elimination wilh and without partial pivoting and digit floating point arithmetic, solve the following system and compare the results_ Show all the intermediate computations_ (2+2)0.1036.1 0.212212 0.7381 0.208151 0.424712 0.9327b) Let AnxnX b be system with unique solution. Compare the nuber of divisions. Iultiplications and additions used in solving the system using Gauss elimination and Gauss Jordan methods Give the exact nlnbers and not the orders

Q3a) Applying Gaussian elimination wilh and without partial pivoting and digit floating point arithmetic, solve the following system and compare the results_ Show all the intermediate computations_ (2+2) 0.1036.1 0.212212 0.7381 0.208151 0.424712 0.9327 b) Let AnxnX b be system with unique solution. Compare the nuber of divisions. Iultiplications and additions used in solving the system using Gauss elimination and Gauss Jordan methods Give the exact nlnbers and not the orders



Answers

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\{\begin{aligned} 3 a+b-c &=0 \\ 2 a+3 b-5 c &=1 \\ a-2 b+3 c &=-4 \end{aligned}\right. $$

Vivid, huge got Jordan elimination method here. Now it's Devlin in this a step vivid light Augmentor metrics. So it will be three minus one minor for three. Two, minus one toe minus eight one to minus three and nine. Now step, do given sit. You first and second room. Then we get Tu minus one to minus eight three minus one minus for 312 minus three and nine. Now step three, we will compute half off our one then minus three off our one plus r two and then minus R one plus r three to clear the first column. Now we get one minus one, divided by do one minus four Joe one divided by two minus 7 15 Joe five. Divided by two minus 4 13 Now step four, we will compute to I go Then half off our toe Plus are one and then minus or five by two R two plus r three. So clear that second column Bigot one Jiro minus 6 11 Joe one minus 14 30. Gero Gero 31 minus 62. No, step five. We will compute won by 31 off our three, then six off our three plus are one and then 14 off our three plus our toe to clear the third column. Then we get one. Gero Gero Joe one. Gero Gero Gero one minus 12 and minus toe so equal to minus one be equal Toto and sequel to minors to ever final answer.

Given the system three A minus b minus four C equals three to a minus B plus two C equals negative aid. An A plus two B minus three ce equals now we want to get this to an augmented matrix first with the coefficients three negative one negative four 2 -12 12 negative three augmented with the constant three negative 89 can do this right away because it's already organized as B's c's equals constants. I'm going to go about this with um the Gaussian elimination with back substitution. So I'm going to get a diagonal local ones. One way to do this is we already have a one in a first position. We can interchange rows one in three. So row two will stay the same and there are many ways to do this. This is not the only way our new row one will be 12 negative 39. In our new row three will be three negative one, negative 43. And now we need zeros below this one. We could take two times row one and subtract wrote to and put that in a neuro too. So row one in three, unchanged in her new road to would be two times one is two minus two is zero, two times two is four minus negative one is five, two times 9, three is negative 6 -2 is negative eight and two times 9 is 18 um minus negative eight is 26. I wanted to zero in Row three, Column 1. So we could take Row three and subtract three times row one For a new row three. So one and 2 unchanged new row three would be three minus three times one at zero, negative one -6, negative four Plus nine is 5. Yeah three minus 27 Is -24. Yeah. Now we want to take we want to one in row two so we can take 1/5 of road to 1/5 of road to would be 01, negative 8/5 and 26 5th. Now we want to zero in row another zero in row three. So we could take seven times row two and add it to row three And put it in row three. So rose one and two unchanged seven times zero plus 00 seven times one is seven plus negative 70, seven times negative eight Is negative. 56/5 um plus five. Right and five is um 25/5 so you'll be -56 5th so Plus 25/5 For 30, negative 31 5th And then we have seven times 26 fits Which is 182, 5th minus 24 is 62/5 and now we're going to row a one in row three in so we can multiply by negative five. 31st of row three, There was one on Tuesday the same and then we get 00 positive one and negative two which means when we translate this. Now that we have a diagonal of ones, we've got the equations, X plus two, y minus three, Z equals nine. Why -8/5? Z equals 26/5 and Z equals -2. So now we can back sol Z equals negative two. Then we have y minus 8/5 times negative two equals 26 5th Or y plus 1650 equals 26/5. subtract 16 to both 16 5th to both sides and 26 -16 is 1050 which is to and if using the first equation, if why is too And Z is -2, we can find X X plus four plus six equals nine or X plus 10 equals nine. So X equals negative one. So our solution is X is negative one, Why is positive two and Z is -2. Mhm.

So we have the metrics. Three minutes to 110 one man. Oh, Swan one minus one, minus four. And also manners too. 13 minos two and five. So 23 minutes. One minus one in minor street. So now we light our one my nose three are two to cutting. You are one and our three plus two are too Got a new are three are four manners who are true to gather new are full So we can see for the first Call it for the first road that he says euro and manners two plus three That is a one wanna know three that is a man of 21 plus tree that is four in the rope Last tells that his hotel her three plus you are true that is a hero in one man is true that ihsaa Manus one's real three Plus you that is a five So my nose to my nose True, that is a man of four And also we have five minutes Eight That is a ministry so further for showy How's your role in three plast food That ihsaa five minus one man is true That is a man of mystery in menace one plus food, that is one my nose. Three plus a That is a five here. So now we light our floor. Last five are three after. Yeah, so gotta new are full and realize our one plus our three to captain You are one we can see for the R one that is a 00 and that it's a plus. So that ico Sewell Stree and zero and I so for the force road that ihsaa five So it's a zero and ministry. So my nose three plus five times five is 25 minutes Three So that is 22 and also one my nose 20. So that ihsaa manos 19 and those So for the last row, we have five minus 15. So now we time the Force Rolley's once Sirte. And, uh, you lied. Ah, course row minus 22 are one to got a new are full. Then we can see our our one No is the road 0103 Tennis 000 and minus 19. So for the last one, we have Ah, my nose 20 Maino 66 So that ihsaa minus 56. What? That it? Matt Men are 76. Sorry. So now we turn the last roll. These miners 1 19 Oh, yeah, that is 00103 one man as well. Um, and swat Manus core to your Romanus 15 minutes, four and ministry. And that is 0001 and four. So now that it's a clear So next up, we just need to get our answer. So rely the secular ral class for Stroh to got new second row. And at the same time, we lied. Ah, certain gro plus yeah, four to got a new are three. And for those Woz one minus one 100 euro minus 15 zero. And for the last one, us minus three plus x team, that is a 30. So now we light our one plus relied are too minus r 12 got Anouar tool. And also we like our three minus five are one to got a new are three. So this is one minus 100 and then those three and just want t Rowe minus one 00 And for this one, ree house 13 miners, 15. That is a man is true and returns are three vis my nose one So we're how? 0100 and two. So we added down. Yeah. So now we only need are two last are 32 got Anouar two, my notes one. So now interchange are one and are to end in interchange are to end our three and we are how 1000 minus one. Jiro Juan's your roads euro True 0012 year old street 00014 So we're how our the answer electors a eco's Who? Manus one b e close to true See Eco's to three and the ICO sue for and we can see this is something

Ve vill huge garden elimination matter here. Now it's Devlin. We will frustrate augmented my tricks. Then we get 31 minus one geo 23 minus 51 one, minus toe three and minus four. Now Steptoe vivid Huge one by three off our one, then minus off to our one plus arto and then minus off our one plus r three to clear their first column. They get one one, divided by three minus one, Divided by three Jiro Joe. Say were invited by three minus starting, divided by 31 geo minus seven, divided by 3 10 divided by three minus four. Now step three. We will compute three divided by seven off our toe, then minus off. One divided by three R tau. Plus are one and then seven, divided by three off are too plus R three, then vegan one. Jiro so divided by seven Joe one minus 13 Divided by seven gentle digital mind, often ma I. And that's one divided by seven three, divided by seven and minus three Now, therefore, we will compute minus off artery, then 13 divided by seven off rt plus arman and then minors off two divided by seven off our three plus Arto Tokyo. Here the column. Then we get one. Gero Gero Gero Want deal, Geo Geo, my name is 16 Entry so minus one. Be able to six and sequel to three is your final answer.


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