5 answers

2Bi The matix 4 H1 hosanrergemkoiveX=1 i0 E 2Selectone 0 8 40 b 3 C 2 0 d 1

Question:

2Bi The matix 4 H1 hosanrergemkoiveX=1 i0 E 2 Selectone 0 8 4 0 b 3 C 2 0 d 1


Answers

(a) 0 (b) 1 (c) 2 (d) 4

.

So this problem we have the following matrix he's equal to for negative 100 0000 00 to negative three. And finally, for the last rose, you go 01 negative too. And we can obtain the care Chrissy equation in terms of Lambda so we can get that using the properties of the determinant of a minus Lambda Chi And we can obtain this as once we do the competition's we will get four minus lambda times Negative Lambda Times two minus lambda times Negative to minus Lambda plus three is equal to zero. So from this factored expression or partially factored expression, we can get Lambda one X equal to four Lambda to equal zero limit of three equals negative one and Lambda four equals one so we can compute the argon vectors as we've been doing in the previous problems. As again as you one is equal to P.

Times 1000 where here pees just ah non zero real number, much like R s in the previous problem. And you too is equal to T times 1400 You three is equal to s Let's just say s the letter after? Doesn't really matter. It's just a non zero real number. It's like a placeholder here, and you four finally is equal to your four is gonna be equal to s times 001 and 1/3. So we can have our fundamental matrix composed from these wagon values and Duggan vectors.

And we have to accept t The fundamental matrix is gonna be given by you to the 40 100 0400 00 each, the negative T t and 00 each that negative t 1/3 you need to the t..

Probably remember seven. Ah, we know that the determinant off a minus. London is equal to zero Francis and servant is for minor zero new to minors, Lunda, and make it good too. Zero negative. 31 minus longer is equal to the road.

So we get none. The plus one. But the blind by London runners who were ex queer does equal to the or so long they quinto and they got the one for an poor. So for Langa is equal to negative one. Um and we know that a m minus Lunda.

But the line where the entity I must have lying my vector being is equal to zero. These gaffes you wear, everyone is equal to zero. And me too. It's equal toe. 33 Ah, the vector is equal to 01 and one.

And who are the basis? Uh, or the ing in for the Indian space Uh, you want so that I mentioned or the one is? Ah. Is what, um who were from days equals before um a minus. No, I'm not Unity. The new vector is equal to zero. Disease gives that to the me too broad.

Three on every three is equal to zero. Singing that there be a vector is were That's a list right into your down. Uh, that on the new vector is different too. Boy One year zero plus s a 03 and two are negative tune we're s And by loans to ah or, um So, uh, Dabney Elector, which is one a zero and zero 03 and negative too, Uh, for the basis, uh, or the I in space e to where? Ah, dimension. Both a two is equal for tomb so we can see the ah, a years No, the sec.

You get, then this matrix A. And first we want to find the characteristic equation which is given by a minus are times I okay. And so is gonna give us, um, you know, eyes the identity matrix, which is ones Diamond Dagnall. So we would be subtracting ours down the diagonal, so they'll be negative. Two minus are 00 for I was to minus are 010 and two minus are Okay, so then we want Teoh take the determinant of this rights that's going to give us negative two minus are times the matrix of negative two minutes Are tu minus r 00 And then for going across the top, we can see that the next thing would be a zero and thirties.

Is you all right? So we would be subtracting zero times, Um, a matrix plus zero times another two by two matrix. But that's just all gonna equals zero. So we can save our time and just go ahead and solve this first piece. Well, of course I copied this matrix. Wrong, right? Should be a negative two.

Here s So we end up getting is a course. Negative. Two minus are times Negative two minus R times Negative two minus R minus zero Uh, which is equal to negative two minus are cubed. Okay, so this means that K is equal too. Negative too.

I'm sorry. R is equal to negative two and K is equal to three, right? It's the power there so that we can use the Caylee Hamilton rule. That theorem rather that says e to the power of a T is gonna be equal to e to the negative to t times he How are a plus two? I de Okay, so this is eat the negative to t times are identity matrix plus a plus two i times t plus Hey, plus two hi square times t squared over two. So first we know I insisted that any matrix, but we want to find what a plus to I is. Okay, so it's gonna take the original A that we have, which is negative.

200 for negative to zero 10 negative too. We're gonna add twos on the diagonal and I dont give us 00000 Oh, I just got excited. Their passion p a zero uh 4001 is your zero. All right, That's not bad looking. And then we will take a plus.

Teoh, I square by square in the Matrix. We just got through in a multiply this thing for zeros by itself, which is not a bad one to do things to all those zeros because we end up with hey, zero metrics. So last but not least, is to put what we just found into our no equation. Right. So we will get, um I'm gonna put these two major cities into into this in their correct places.

So we end up with either the negative, too. T times have the identity matrix plus tee times. This mostly, but not all zero matrix him and plus t squared over there to times are zero matrix. So if we add all these together, we're, um, e to the negative to t. Because I don't need these parentheses anymore.

Um, you know, for this first row first column, we'd say one plus t time zero plus t squared over two times zero seconds of just being one. Uh, And for this next spot, we would say zero plus t time zero plus t squared over two times zero since Columbus zero. Okay, the next one will be zero You the same thing for the next row. Because that's for tea. One of the same thing for the next row.

Because this t zero and one, yes, that's not a terrible looking matrix. And because it's rather simply my go ahead and even distribute that Eat the negative to t. Okay, and then that solves it. So, part B. It is this matrix..

Yes. Okay, so here we have a four by four as a matrix, and it has the falling increased 0100 one negative. 100 and four zeros. It's left corner here and then 01 negative. Two and four.

Now we can use Matt Love again and use I command to find the alien values and argon vectors of a and they're all gonna be consisting of decimal entries. Um, real numbers. So they're required fundamental matrix that we want. It's gonna be function of tea, as we can see on the left hand side. And it's gonna have the first entry, for instance, as 0.8507 e the 0.698 t and the other entries are gonna be following it.

Let's say this. This is another decimal entry and this third entries 060 And the next row is a decimal entry. So on zero and zero, and we noticed that the zeros structure that is present here and here is preserved by or this year structure is present in a on this side, and this one is preserved in this matrix because the last four entries are decimal entries again. Kind of like what we see here. We only have zeros, continuous zeroes or blocks of zeros in the top, right and bottom left..

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