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0 -3 A = [- 2 7 = 7 3 6 _...

Question

0 -3 A = [- 2 7 = 7 3 6 _

0 -3 A = [- 2 7 = 7 3 6 _



Answers

$$a=-7 / 2, \quad b=-1 / 2, \quad x_{0}=-3$$

Here we have the Matrix A given by this with the entries negative. 706 05 year old and six zero Number two. And so we have the characteristic equation for this problem can be written as follows Lambda minus five squared times. Lambda plus 10 is equal to zero. So we have that lambda one coming to come on. Three people's five color five common *** 10. So now if we choose Lamb dick them tow one equal to five that we can find the Egan Vector for the first wagon value using the same steps that we did in the previous problems. As given by s time, 010 Where s is a free non zero rial number and Lambda two equals five. So that's the other repeated Aiken value. It's going to give us you two equals as times 102 Thank you. And finally landed. Three equals negative 10. It will give us You three is equal to s times native to zero and one so we can write the general solution as except T is equal to see one eat the five t times you one just given above plus C two times each of the five. T You too. It plus C three each of negative 10 t times you three. And for the sake of being specific, we can take s equal a one off these cases. Since any s works, we can take s equals one to give us the specific version of you on your twin. You three. So we'll have, for example, here you'd have zero 10 And so here we have 102 And so that's our final sleep.

So here is our interval right here. It's from negative, uh, 7/2. Teoh Negative three. Ah, Or to negative 1/2 and negative three have is the x not value. So as long as, um, our value of delta is somewhere in this interval, but is not negative 1.5 or Okay, so we could say that negative two or negative one is our answer.

Okay, so here were given the Matrix A is equal to three negative, too. Zero and three. Okay, So first I want to find the characters to the equation. And we do that by taking a minus r times the identity matrix. Okay, so that's gonna be equal to three negative to zero three minus since me ours on the diagonal. And so that gives us three minus are negative to zero three minus are Okay, so now we take the determinant. And so the determinant Ah, a minus are times I is gonna be equal to three. Minus are times three minus are minus zero times negative. Teoh, uh, which is just equal to three minus are squared. And so one of the first things we were supposed to do in the prompt is to show that ah, this a minus are times I to the power of K is equal to zero for some number are right into, um r k then is gonna be this too. So you're supposed to show that a minus. R I The power of K is equal to zero. Okay, So K is equal to two and we move on and use the Kayleigh Hamilton, the're, um, and the Killi Hamilton here, um, tells us that e to the power of a T is equal to e to the power of are tee times e to the part of a minus. R i t he also, from here, we can see that are has to be equal to three. And it's squares. That should be, um, two values that would satisfy this. But the only option is gonna be three. So it's like three or three. Um okay. And so then this is e to the three t times e a minus three. I t okay. Also, that is equal to e to the three tee times I plus a minus three. I t. Okay, so we have values for all these matrices. This is gonna be e to the three t times are identity matrix is 1001 for us. Bring it a which was three. Native to cereal, three minus three. I was gonna be three other diagonals and multiply that by t. Okay, so that we consol of these additions attractions. So we get e to the three t times. Start with the second Paris that in their brackets. Yes, this is gonna be able to tee times Season three miles 30 negative. Two minus zero. It's negative to zero minus zero than three months. Three. Okay, so the next step here. And so we would take this tea and multiply everything in there. So of course, we're gonna get a bunch of zeros. And then what would be a negative to t in the right hand corner? Someone to skip that step on and we end up with is just he times the matrix one negative to t zero and one. Okay, well, now you have to multiply that He across everything, right? So our final matrix ends up being either the three t negative to t either the three t zero and e power of three t. Okay. And something that should be the final answer.

Yeah. In order to solve this equation, since it's already written in factored form, we're just going to take each one of our factors and set them equal to zero. So three a minus 10 equals zero And to a -7 equals zero. Now we just need to solve each one of these equations. So for the first one will start by adding tin to both sides And then we can divide both sides by three. So we end up with a equals 10/3. For our first solution, then we'll do the same thing with our second equation. We'll start by adding seven to both sides And then divide by two. So a equals 7/2 is our second solution. So now all we need to do is check our answers to check those. We're gonna take the A values the values that we got and we're gonna plug those in for the A values in our original equation. So starting with 10-3, we have three times 10/3 minus 10 Times two times 10/3 minus seven yeah equals zero. So when we multiply for this first fraction, the threes are going to cancel out and leave us with 10 -10 which is zero. And then for the second factor, when we multiply this we're going to get 20/3 -7, which uh we can go ahead and combine those but ultimately we don't really have to because we already know that whatever this number is, it's gonna be multiplied by zero. So any time we multiply something by zero, no matter what this number is, the product will end up being zero. So we can say that zero equals zero. And our first solution, 10/3 is a good solution. Now, we're going to do the same thing with 7/2, we're going to plug that in for both of our values. So we have three times 7/2 minus 10 times two times 7/2 minus seven equals zero. So for the first factor when we multiply this we get 21: 2 -10. Yeah. And for the second factor, once again the twos will cancel out. And that leaves us with 7 -7 which is zero. So once again, even though we haven't finished adding these numbers together, we really don't have to because we're multiplying it by zero and anything multiplied by zero is always zero. So again our solution checks out so we have to Solutions to this problem a equals 10/3 And 7/2. Yeah.


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