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

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

Algebra

Chapter 2

Matrix Algebra

Section 2

The Inverse of a Matrix

Introduction to Matrices

Campbell University

Harvey Mudd College

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

01:20

Suppose $A$ and $B$ are $n…

03:37

Let $A=\left[\begin{array}…

01:37

Show that if $A B$ is inve…

00:18

Fill in the blanks.If …

02:00

Suppose $(B-C) D=0,$ where…

01:18

If $A$ is a $2 \times 2$ m…

00:50

Suppose that $A, B,$ and $…

01:43

Use determinants to prove …

01:44

Suppose $P$ is invertible …

03:51

Use the equivalence of (a)…

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're going to make a few assumptions here. First, we'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're square? Um well, one reason that it helps if they'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't do it the other way. I can't say that four by two and three by four I can'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'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'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'm multiplying this on the right hand side, I need to do the same here. I can'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

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