solve-the-equation-a-bb-c-for-a-assuming-that-a-b-and-c-are-square-and-b-is-invertible

Name: solve the equation a bb c for a assuming that a b and c are square and b is invertible
Uploaded: 2020-11-18
Duration: 2 min 2 s
Channel: Numerade
Description: Solve the equation $A B=B C$ for $A,$ assuming that $A, B,$ and $C$ are square and $B$ is invertible.

Question

Solve the equation $A B=B C$ for $A,$ assuming that $A, B,$ and $C$ are square and $B$ is invertible.

Jacquelyn Trost · Accepted Answer

for this problem. We want to solve the equation. A Times B equals B times C. We want to solve this for a now we&#x27;re going to make a few assumptions here. First, we&#x27;re going to assume that a, B and C are square and that b is in vertebral Now, why is this important? Why do we care if they&#x27;re square? Um well, one reason that it helps if they&#x27;re square is that I can multiply them together. Remember when we multiply matrices, if I have Yeah, a three by four. I could multiply that by a four by two because the rows and columns of one match each other. But I can&#x27;t do it the other way. I can&#x27;t say that four by two and three by four I can&#x27;t multiply these together So by having them square multiplication is possible in any direction. Such just kinda makes us a little bit nicer. We don&#x27;t have to worry about the sizes. Everything multiplies by everything else, right? And also be being in vertebral. We are going to need to take the inverse of B to do this so we know that we physically can. Okay, so we have a times B equals B times C. Now, how do we solve this for a Well, I need to get rid of that. Be so what we&#x27;re going to Dio is have a times b times the inverse of B. Now, remember, with matrix multiplication order is very important. Since I&#x27;m multiplying this on the right hand side, I need to do the same here. I can&#x27;t just say be times inverse of B and put them together on the right hand side. I have to put this in the same position. So what does that give me on the left hand side? Be times the universe of B is just is the identity matrix, so that can go away on the right hand side. I do need to show all three b times C times the inverse of B

Angelo Rendina · Answer

So let&#x27;s go see a times B and seizing veritable So C minus one exists. So what we do here is we multiply both sides from the right by the inverse of C So we have C see in verse equal a B see embers. But now see, see in verse is by definition of inverse get entity metrics And on the inside we have a me CM birth Now about the associative property or matrix multiplication. We can put the brackets here and this formula is telling us that BC in verse he&#x27;s the inverse away so easy in vegetable.

Jack Chen · Answer

Okay, So Okay, so in this program, if a caused to a block can be ah petitioning into a blocks of matrices B 00 on See now first direction If e and C our convertible that muse, beamers and the seamers exist, then we can directly construct yet the a inverse to be immersed seamers and 00 So if it before because dropped another men traces like this these metrics is thievery is okay because we can verify that eight times a humorous because toe humorous times a equals two and I grieved images. So this is one direction. So we label is by one for the other direction if a it&#x27;s incredible. So that means a universe exist and the Immers has the same dimension is a them use. We can petition a humorous by X y z w in the X and the B has the same size and w and the sea have the same size. And since we petitioned both a numerous conformity so we can do the multiplication eight times a humorous equals to the 00 c times x y z tabouk that gives us p x b y Sesay, see Bablu that way. We already know that a inverse eight times a humorous equals two a night in the Matrix And, uh, we can petition abuse. I think it metrics by this. And by comparing both metric mentions this we know that be X equals dough I the CW because toe I So this is one direction and we can also we can do the same thing for a m Rose time say, sees a MERS 10 say&#x27;s also we had any matrix It gives us x times p equals toe I and the tablet times see caused by. So our conclusion is B and the sea uh, also in vertebral because this is exactly the definition off. You&#x27;re worth ability in the night proof this statement if they can be petitioned like this. So a is irritable if, and only if both being a c I multiple

Runpeng Li · Answer

So in this problem, we&#x27;re going to show if a b a Times B is in veritable Hubert. Oh, then he is in vertical. Okay, So in order to show this by our convertible matrix B room, we know that Bye. I am t. There exists a matrix c Such that. See, time&#x27;s a b, his identity. So that is equivalent to guess the matrix multiplication, he said. So associative associative. So we have see, Time&#x27;s a time speedy his identity. So Dad means See, Time&#x27;s a is the inverse. Uh, so we conclude that be is convertible and we are done.

Joshua Eastwood · Answer

So this question says, If a is in vertebral, the solution to this equation a X equals B is wider. That&#x27;s a fill in the blank, and then we&#x27;re gonna fill it in by saying X is a inverse times be.

Angelo Rendina · Answer

so we have be minus C that multiplies the equal to zero. So since the is in vertebral, we can multiply from the right by the inverse of the so right B minus c the and then the inverse. And that&#x27;s equal to zero times Demers. Now the product the D in verse is just it. Entity metrics. So this becomes B minus c that multiplies the again. Did he is equal to now Any metrics multiplied by 00 So on the right hand side, we just have zero and now must be bribed by the identity matrix does nothing. So on the left hand side we have B minus c and on the right inside we have zero Now we some ah si on both sides. So that&#x27;s right, B minus c and then we add plus C on the right. Inside we have zero Blasi Well now minus C plus C is just zero. So are we on the left inside? We just have B and on the right inside something zero does nothing. So we&#x27;re just left with C, which is what we wanted

Problem

Suppose $P$ is invertible and $A=P B P^{-1} .$ So…

Problem 17 Easy Difficulty

Solve the equation $A B=B C$ for $A,$ assuming that $A, B,$ and $C$ are square and $B$ is invertible.

Answer

inverse of $A B C$ is $C ^ { - 1 } B ^ { - 1 } A ^ { - 1 }$

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

Linear Algebra and Its Applications

Anna Marie V.

Kayleah T.

Maria G.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

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inverse of $A B C$ is $C ^ { - 1 } B ^ { - 1 } A ^ { - 1 }$

Linear Algebra and Its Applications

Anna Marie V.

Kayleah T.

Maria G.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

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

Anna Marie V.

Kayleah T.

Maria G.

Michael J.

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

Linear Algebra and Its Applications