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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.]$\left[\begin{array}{ll}{A} & {B} \\ {C} & {0}\end{array}\right]\left[\begin{array}{ll}{I} & {0} \\ {X} & {Y}\end{array}\right]=\left[\begin{array}{ll}{0} & {I} \\ {Z} & {0}\end{array}\right]$

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

Algebra

Chapter 2

Matrix Algebra

Section 4

Partitioned Matrices

Introduction to Matrices

McMaster University

Harvey Mudd College

University of Michigan - Ann Arbor

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Okay, So we're doing some matrix algebra here where I'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's the identity. Um, I'm not even gonna look at the right side. I'm gonna write it down. Just Syria, the Identity and Z and zero. So as a reminder when you multiply because that's what's happening here is you do rose by columns. So it's eight times the identity, which is just a plus be times X. So that's how you do your matrix multiplication. And it has to equal row one column one over here, which is zero. Um, and that's what I'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'll do all four of them. But if I moved down there Row two column one. That'll equals e So as you go through that we have C times the identity, which is just see, I don'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'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'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.

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