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Suppose the first two columns, by and $\mathbf{b}_{2},$ of $B$ are equal. What can you say about the columns of $A B$ (if $A B$ is defined)? Why?

if the first two columns of matrix $B$ are equal then the columns of $A B$ are equal.

Algebra

Chapter 2

Matrix Algebra

Section 1

Matrix Operations

Introduction to Matrices

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01:32

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Suppose the second column …

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Explain why the number of …

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What must be true about th…

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Show that if the columns o…

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If $A=\left[\begin{array}{…

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Writing Suppose $A$ is a $…

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CAPSTONE Let matrices $A$ …

Okay, so this question is asking us. Suppose the 1st 2 columns of B R. The equal are the same. What can you say about the columns of a Times B? And we're going to use the fact that each column of a Times B is a linear combination of the columns of a using the corresponding column of B as weights. So what that means is, let's say that a be sub one is the first column of the Matrix a B, and that's gonna be equal to first column of a plus second column of a plus third column of a and so on using the entries of the first call of a B. So be 11 be to one be 31 and so on. So the the column vectors of a here are acting as sort of the variables that we're including in our linear combination, and then the the weights of the linear combination are the entries of the corresponding column of ah of be, like notice how this column, the number representing the column matches the column where we're working with the whole way through. And so the 2nd 1 will be the same. So the second column of a B will be equal to well, well, Instead, we'll be using the second column entries of Be So this will be Be one, too. Ah, a one plus B 22 a two plus B +32 a three. Right and so on. But if the first column of B is equal to the second call, maybe then, um, be I One is equal to B I to is equal to just be so. I will just call it. We'll name it will name them all be so bye for I want to end if he is an n by n majors or as an collins at least. But that means then that these two mate, uh, the two columns a B one in 82 are equal, right? Let me just write them both out, all right? They'll just be able to be one times the first column Vector of a plus B two times The first column vector of a second column vector of a plus B three times the third column vector of a and so on. So these two columns will be equal, and that's what we can say. But the columns of baby

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