find-formulas-for-x-y-and-z-in-terms-of-a-b-and-c-and-justify-your-calculations-in-some-cases-you-ma

Question

Find formulas for X, Y, and Z in terms of A, B, and C, and justify your calculations. In some cases, you may need to make assumptions about the size of a matrix in order to produce a formula. [Hint: Compute the product on the left, and set it equal to the right side.]
$\left[\begin{array}{ll}{A} & {B} \ {C} & {0}\end{array}ight]\left[\begin{array}{ll}{I} & {0} \ {X} & {Y}\end{array}ight]=\left[\begin{array}{ll}{0} & {I} \ {Z} & {0}\end{array}ight]$

Gregory Higby · Accepted Answer

Okay, So we&#x27;re doing some matrix algebra here where I&#x27;m just going to compute the multiplication on the left side of the equation. And remember that you do rose by columns going to say that that&#x27;s the identity. Um, I&#x27;m not even gonna look at the right side. I&#x27;m gonna write it down. Just Syria, the Identity and Z and zero. So as a reminder when you multiply because that&#x27;s what&#x27;s happening here is you do rose by columns. So it&#x27;s eight times the identity, which is just a plus be times X. So that&#x27;s how you do your matrix multiplication. And it has to equal row one column one over here, which is zero. Um, and that&#x27;s what I&#x27;m gonna continue to do as I move on to the next column. So eight times zero. Well, thats zero. Anything times 00 So that was a waste of time. Um, plus B y is equal to row one column one. Sorry. Row one column to which is equal to one right here. Um, yeah, and then all I really need. Next, I&#x27;ll do all four of them. But if I moved down there Row two column one. That&#x27;ll equals e So as you go through that we have C times the identity, which is just see, I don&#x27;t need to write down the identity plus zero times X just like here. Zero times anything is zero is equal to Z. I think I can stop there. But because the last one I wrote two column two is pretty redundant because anything times 00 plus zero times anything is zero is equal to zero. Yeah, that was redundant. 00 equals zero. So these are what we need to look for. This one&#x27;s already solved. That Z is equal to see. Um, now this one, all I would do is try to get rid of B to sell for why? And the order does matter that you multiply left to right. So you do the inverse on the left side. So why is equal to Well, anything times the identity is itself. So why is equal to be inverse? And then, uh, then go up to the top where you would want to subtract the A matrix over a value over first and zero minus anything is that thing. And then dio be inverse of that, um and ah, Ah, unique property here is that you can have that negative in front. So negative. Be inverse A, um And it&#x27;s the same logic is what I did with. Why here, as you do, be inverse to cancel that out. Um, yeah. So this would give us the answer.

Runpeng Li · Answer

so. Four problem. Six. Consider to partition matrices x zero Why&#x27;s he times a zero BC So we need to find formulas for X y Z even turn themselves ABC so that the first thing we need to do is just find product of these two matrices. So let&#x27;s do this. Let&#x27;s be issue papers X times, eh? Plus zero times visas just a zero and X times zero plus zero times to see some time. See, So that&#x27;s just zero. And why time&#x27;s a Why time Say us z times be and the time see And this is equal Thio I zero and zero I So that means X times, eh We&#x27;ll be I so that gives axe to be in bursts of a Excuse me. Okay. And for this entry here we have ze time. See? It was I so that means c should be the reversals. Sorry, z should be the numbers up, See? And by the third entry here, we know why times a us the times be will be zero. So that is we put a z times be from life inside to the right hand side. Then we know why time&#x27;s a will be negative z times peak. And we know Z is seeing verse facilities active. See inverse times people. And that gives the Matrix. Why? So that is negative. Z Sorry, connective. See in verse B times a Evers.

Runpeng Li · Answer

All right, so for problem eight, we have a B zero i times X. Why? Z 001 Sorry, I And again, we need to find the formulas for X y Z in term self ABC. So, uh, first do the matrix modification, so we have a X plus zero times. Be so it&#x27;s just a general and a y and ese eight times, Steve plus B times, eyes would just be okay. And for the second row, we have, um zero first and again zero. And this will be a zero times C and zero. So it&#x27;s why squares just why? Sorry, I square just I and this is equal to buy over assumption. We know this is equal to ay 00 00 I Okay. And now, by the first entry for century in the first row, we know a taxi a axes equal to ay. So acts can be reading nest a verse and a Y is equal to zero. So that implies why is zero And also we know ese easy plus B is zero must be is zero. So that implies, ese. It&#x27;s negative. B and Z, therefore see will be active. A inverse times be and this is sour

Jack Chen · Answer

Okay, So in this purpose left inside in this equation, you used to like 00 y zero. I not tried by a Z 00 b i and off this summit seven matrixes are petitioned conform obl e So we can just directly use the matrix multiplication. So we have X a x z. So the result will be a two by two matrix and by so this is 11 site in the right hand side of this equation because toe and identity matrix, that means X a closed toe I, which gives us X equals toe a evers and that we know that&#x27;s why, um why a class B equals 20 that gives us why he caused to minus Pete times a humors and the four Z We know that xz cause zero. That means he has to be zero. So this is the solutions. Are these other solutions to this program

Thomas Emment · Answer

Okay, So you want to write this out as a matrix screen? So first off, that&#x27;s right. Are variable matrix. We&#x27;ve got X. Why on is that on? This is gonna be equal. Thio answers heads who have one for two. Next up. We just need to pick up all the cut out all the coefficients in front. So our first road here Well, that&#x27;s gonna be what? It&#x27;s gonna be tough. Firaxis who want relaxes their sort of 1 to 1 the next. That is gonna be the wise 11 minus one. And they will use that one minus one. Why does three okay? This is how the major to look.

Lucas Finney · Answer

All right, so here we are, given a matrix a 100 x 10 zed. Why one? And we&#x27;re told that a squared plus 0 +00 negative. +1000 Negative +10 is equal to the three by three identity matrix. So we need to figure out what x y and Zed are. So the way that we can do that is first, let&#x27;s figure out what a squared is. So a squared is gonna be a times a which is going to be I&#x27;m just gonna copy these guys down. Uh, that pasty. Yes. There we go. So only real way to figure out what the square of the Matrix is, unless it&#x27;s in a very particular form is to do the matrix multiplication by hand. So and the i j element of our product here is going to be the dot product of Roe I from this guy on the left with call on J in this guy on the right. So element 11 21 times one plus x time. Zero places zed time. Zero. That&#x27;s just gonna be one element. 12 Yeah. Sorry. Actually, I&#x27;m gonna go to one zero times one plus one time. Zero plus white times zero Gonna be 0310 times, one plus zero times you&#x27;re a post one time 00 12 is going to be one times x plus x times one plus that time zero. So that&#x27;s going to be two X up there element to 20 times x plus one times one plus Why time zero Going to be positive. One element 3 to 20 0 times x plus zero times one plus one time. Zero Just gonna be zero Element 13 is going to be one time said, plus X times Why plus zed times one. So we get to, um So, yes, that would be to Zed. Plus X Y is our element 13 there element to three go t zero times said, Plus one times why. That&#x27;s why times one that&#x27;s going to be, too. Why and then element 330 times ed plus zero times y plus one times one is one. So we now need to solve the equation. One second here. So we had a square. Yeah, I&#x27;ll just copy and paste these guys from above. So the way that we can look at This problem now is that we actually have ah bunch of little equations to use to solve for our three unknowns the X Y and said because one thing that we needs toe have If so, we can check a few things. First of all, that one plus zero is going to give us the one that we need there. That&#x27;s OK. One plus zero is going to give us the one that we need there. And one plus zero is going to give us the one that we need there. So what? This what is useful to us here is that if we look okay, we&#x27;d have to have two x minus one equals zero two X minus one equals zero. Tells us then that two X has to equal one or X as equal 1/2 then two. Why minus one equals zero similarly tells us that why has to equal 1/2 then lastly, we have to zed plus x times y has two equal zero or sorry to zed plus x times Y plus zero has to equal zero. So that tells us, lastly, that to zed is equal to negative. X times Why, but we know what X and y are A to Zed equal. Negative 1/2 times 1/2 is going to be 1/4 Tuesday is equal to 1/4 or a negative 1/4 and said has to equal negative 18

Problem

Find formulas for X, Y, and Z in terms of A, B, a…

Problem 5 Easy Difficulty

Answer

$X = - B ^ { - 1 } A , Y = B ^ { - 1 } , Z = C$

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Alayna H.

Kayleah T.

Maria G.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

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$X = - B ^ { - 1 } A , Y = B ^ { - 1 } , Z = C$

Linear Algebra and Its Applications

Alayna H.

Kayleah T.

Maria G.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Alayna H.

Kayleah T.

Maria G.

Kristen K.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications