Hello there. So for this exercise we have these two systems of uh these two linear systems. But basically you can observe this response to the homogeneous system and this to the non homogeneous system. Yeah. So then the first part we need to find the general solution for the homogeneous system. So let's represent this as an extended matrix. So that means taking these matrix here. 000 Okay, but look, we have something happened here. The first the third bro is equal to minus the first row and the second row is two times the first row. So technically these two roads will eliminate by doing the following operations. Road operations. So if we take our two and then we say that our two minus two times R one and R three will be our three plus are one. By doing those operations. We obtain the following. So just let me copy this would write everything. So these two rows will become zeros 000000. So we need to find the solution for this system. You can see by the matrix that we have only one pilots. So that means that we have to free variables In this case X two and X three. So the solution will be dependent of X two and X three. So let's give some values to X two. Let's say that extra pickles 23 annex three. Uh is equals two. S. where T&S. Are real numbers. And having this in mind then we can find a solution for X one and X one is equal to one third times minus two. S. T plus S. Okay, so from this we obtained the general solution for the system and it's the vector that depends. Two variables. T. M. S given by one third. I'm going to take one third as a common fracture. And then we have here minus two T plus s minus two plus S three times T and three times as. So this is the solution. The general solution for the homogeneous system and actually we can write it better. And is that these solution? It's called X. H. For homogeneous surgeon. So X H t. S Can be written as the sum of two vectors. So will be T thirds time. Is the victor -230 Plus as over three as 3rd. 103 Great. So we have this and well we can replace this by some alter a constant. It doesn't matter. These are just as killer. So we can just replace right tms is the same solution. So this corresponds to the Alma genius solution of this system. Now we need to verify for the part B A particular solution. So we need to verify that the following vector 101 is a solution for the non homogeneous system. So that means to this system here so we copy this. Okay, so we need to verify that this vector here is solution of this. So let's check here. We're going to put the vector 101. So the multiplication of this will give us three, 6 -2 -3 Plus one. And you can observe that there is a result in 2: 4 -2. So this vector here XP is a particular solution of the system now. So for the part C we need to use these results, the particular solution and the homogeneous solution to give a general solution for the system. And that means that the general solution, let's say x g t s will be greeting us. The homogeneous solution plus the particular solution where the particular solution is the cost 101 and the homogeneous solution depends on two variables, T times -2, plus S one 03 So we just need to put the things together. So the general solution X G depends on T m s will be T times the vector -2, plus S. The victory 103 And the particular solution that is the victor 101 And this is the general solution of this uh in a homogeneous system. No, we need to verify that actually this is true. So to verify that we need to solve this system, the in a homogeneous system. So we need to find a solution of this. So to do that we need to consider the extended matrix and that means taking this matrix here two, four minus two. Okay, again, we're going to perform the same operations as in the homogeneous case. So we take our two will be Our 2 -2 times the first row and the third row will be just some of the earth, the 3rd row and the first road. And doing that, we obtain the following. So these are become zeros and these 20 and zero. Okay. So we think this from this expression we again we have to free variables X two and X three. And the general solution of this system is also dependent of T. N. S. And it's Graydon us T times -230 plus S 103 The particular solution in this case is given by two thirds zero. Um So we need to verify these. These these also for response, we can generate the particular solution. Consider in C. In B. Sorry. Uh We can consider so. XP can be greeting us some value of T. And s. Of these particulars. That is clear. Well it's not directly clear but we need to compare this so we need to generate the filming vector 101 should be equal to this part here to be some of these vectors. Yeah. So the first thing just pay inspection, you can observe the T. Should be equal to zero because we don't have any value here. Right? So T should be equal to zero. That's the first thing, this zero and we need a number one here. So that means that S should be equal to one third if S is equal to one third on the right hand side. We have the full we have the picture 1 3rd 01 plus the vector 2/300. And you can see that this is the battery 101 So there's a particular solution of deal in um a genius system.