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Write a matrix equation that determines the loop currents. [M] If MATLAB or another matrix program is available, solve the system for the loop currents.

(a) $\left[\begin{array}{cccc}{11} & {-5} & {0} & {0} \\ {-5} & {10} & {-1} & {0} \\ {0} & {-1} & {9} & {-2} \\ {0} & {0} & {-2} & {10}\end{array}\right]\left[\begin{array}{c}{I_{1}} \\ {I_{2}} \\ {I_{3}} \\ {I_{4}}\end{array}\right]=\left[\begin{array}{c}{50} \\ {-40} \\ {30} \\ {-30}\end{array}\right]$(b) $\left[\begin{array}{c}{3.68056} \\ {-1.90278} \\ {2.56944} \\ {-2.48611}\end{array}\right]$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 10

Linear Models in Business, Science, and Engineering

Introduction to Matrices

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problem. Five, We want to construct a metrics that determines the loop currents. I want I to I three and I four and it's directions are given in the problem. In this circuit, we will take each group and a ploy. Kershaw Slope which says that Sigma ir in the loop equals V. Is this her shift slope? It's take the first loop. The first loop we have in this resistance. We have ion in this direction on we have I want multiply it by our on. We have here all the one minus I to because we have the current from my two goes in this direction. Or we can write it as to arrows we have here I wan are you want you have here I one and another by two. And we have here Iowa while this voltage goes in this direction from negative to positive from the negative to bust. Then for the first loop, applying ir equals V. We have I one is multiplied boy on the resistance which is four plus one plus one plus five. And we have Are you two in the opposite direction minus I two multiplied boy for you. These old equals 20. This is from the first loop. From the second loop we have, we can continue in the same color. We have a two in this direction. We have a two then we have a two and we have a three. From this loop I to 43 years I to it is eight and the direction off the voltage. We have missed this voltage in the first loop. It goes in this direction. Then it's 20 and 30 20 plus 30. From the second loop we have the same voltage and it goes in this direction in the loop. And we have another voltage which goes in this direction. Are you one? I two is multiplied by all the currents in the group or the resistance in the group which is one plus five plus three plus one minus. We have all you want multiplied by five and we have minus I Sorry, multiplied by all this old equals the voltage. We have two voltages. We have 30 and one, 30 and 10. We have 30 votes and 10 votes and the two has a negative direction. That relative to the direction of eye to I two revolves in this direction and 30 volts revolves in this direction. This is V to this is we and we have 10 volt revolves in this direction. Then we have minus 30 minus 10. You can change the color for the third group. We have I three in this direction. I three i three and I three and we have I fought in this direction and the direction off 10 votes in this direction and 20 vote from negative to positive in this direction. Then we have I three multiplied by all the resistance in the loop two plus one plus four lost to minus. Oh yes, three. Multiply it by one Sorry minus I to multiply it by one minus I food multiplied by two These old equals the voltage. We have two voltages and the two voltages has had the same direction which is thin the last 20 for the fourth loop. We have iPhone and we have iPhone, iPhone and I for and we have the voltage is this direction and we have This is the same direction off the 20 bowl. Then we have I four multiplied by all the resistance in the loop three plus two plus one plus four minus. We have I three multiplied by two All equals minus 20 minus 10. Because the voltage as a direction opposite to the direction off the current in the room. Using these these four equations, we can construct the metrics. The first the column in the metrics. The first two column has the coefficients off. Um, in all in all equations. Ah, coefficient off. Ion here is five plus five plus one is 11. The coefficient of I wan here is minus five. The coefficient of ion here is zero. And here is zero. Then we have the second column. Is the confessions off? I too In all the equations we have minus five. Then we have six plus 39 We have 10 here we have here minus one. And finally we have zero. The third column is the coefficients off I three we have here three plus 47 plus 29 Sorry. The coefficients of I three from the first equation. We don't have I three in the first equation. Second equation, we have minus one. The third equation we have three plus four equals seven plus two equals nine and we have finally minus two. Yeah, in the whole equation. The fourth column is the confessions off I four iPhone. We don't have iPhone in the 1st and 2nd equations. The third equation, we have minus two. And the fourth equation. We have three plus two, which is five plus 16 plus 4 10. And this is the metrics. And we have the Fifth Column is the right hand side for all the equations we have 50 minus 40. 30 on. Finally. We have we have minus 30. This is augmented metrics. Or this is if we want toe, write it in a metrics equation. In the equation, we have these metrics nine minus 200 minus 2 10 multiplied by the four currents. I want I to I three and I four equals the right hand side or the voltages with 50 minus 40 30 and minus 30. And using software, we can enter this in matter, for example, to get rid used an equivalent metrics, equivalent metrics. That is as some sort of the of the metrics. 1000 3.68 And we have 0100 minus 1.9 zero and we have zero 010 two points, 57 And finally we have 0001 minus two point for mine. And this from this form we can see that we have the four currents. I one equals 3.6. It I'm better I to equals minus 1.9 bears, which means it's in the other direction. Then they assumed in the problem. I three equals 2.57 bears. Finally, I four equals minus 2.49 Amber's and this is the final answer off our proper

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