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If $a, b,$ and $c eq 0$ are real numbers with $a c=b c,$ then $a=b$ Does this same property hold for matrices? In other words, if $A, B,$ and $C,$ are matrices and...

Question

If $a, b,$ and $c eq 0$ are real numbers with $a c=b c,$ then $a=b$ Does this same property hold for matrices? In other words, if $A, B,$ and $C,$ are matrices and $A C=B C,$ must $A=B ?$

If $a, b,$ and $c \neq 0$ are real numbers with $a c=b c,$ then $a=b$ Does this same property hold for matrices? In other words, if $A, B,$ and $C,$ are matrices and $A C=B C,$ must $A=B ?$



Answers

Suppose that $ a \neq 0 $.
(a) If $ a \cdot b = a \cdot c $, does it follow that $ b = c $?
(b) If $ a \times b = a \times c $, does it follow that $ b = c $?
(c) If $ a \cdot b = a \cdot c $ and $ a \times b = a \times c $, does it follow that $ b = c $?

Hung eyes in this video. I'm going to be entering this question down here. If a Times B majors touches be equal zero, does that mean that matrix A or major speed has to be zero in regular multiplication? We know that if a times b equal zero, we know that either eh house equals zero or B has equals here or both of these values can be zero. However, with matrices, we're going to prove that this is not necessarily true. By doing this problem right here a times B and the first step to doing so, what is to make sure that a has the same number of columns as B has rose two and two. So we're good to go here. We're gonna take the first number of each of these two things. We just circled and multiply themselves three times. Three. Oh, sorry. Three times one and then plus three times negative one. They were going to do the same with RO one of matrix a Border news column to of Matrix B. So it would be three times negative. One plus three times one. And now we're going to take the second row of a and multiplied by the first column B just like we did up of so four times one of plus four times negative one. And then again, we're going to use the second column of B over on this side. So four times I get one plus four times one, and in doing so, we've created a new matrix called Matrix A B, and we're going to finish this by simplifying matrix a B. So going to add these terms together? So three plus negative three is going to be zero negative three plus 30 four plus in negative four and negative four plus 40 So this matrix is equal to a B and also equal to zero, and therefore we can see that just because matrix A B equals zero does not necessarily mean that Matrix A or Matrix B equals zero. Feel free guys to pause and either re players rewinds and Parsons video if you need extra help. Thank you guys so much for watching, and I really hope you got something from this

All right. So for this one, we're gonna say that no party is a big no. And as though his partied part seed also enough.

Okay, give me to make. Seems to a be a little for a Be on, then be a and have a look if they are the same. So that rose from the first Matrix columns. The 2nd 1 some perfect. This magnification is the rows of a against common big Much monetary for for his three is gonna be 12. So gonna have 16. 16 in this exhibition That's really sank on to find a seven head. Martine's one is minus two full against six is going to be plus 24. So that with 22 Not really. First car one minus two minus 23 lots off three is gonna be plus nine. So that's just great to be seven secondaries and come here with 112 lots of six is gonna be plus 18. So he left in 19. Okay, so now we're taking basically Rose our biggest calling, my 21 against miles to one. So that's going to be much of his for want of. One is once we're gonna find that. Okay. Already conceded thieves are different in General Macy's investor like me. Best in each other or I destination something. Make sure she's changing positions on a modification will give you different major cities apart from its there. This is, um Well, if that you have it, this matrix. Okay, so a second would be second row first column off eight. Finest vision here. We've got 36 gets minus 21 to 3 to minus two. Is my six on that? We've got six against one, which is six. You have zero that. It's a fast rate secondary. First call. It said, first. Very second. Forget best. Very that, Robert. That first very Chris column. Seven first sand comments. You got my team. Also, before you have the hands mice ate and went up to take us three such could be to minus five five, then Second column Feels the four is 12. See, close to 3 18 So we're gonna have a 30 there. Okay, so these teammates seas are different, okay?

S so we have A and B which are both two by two matrices on We have a plus B squared. So a plus B squared is equal to a plus B times a plus B And because they're both two by two matrices, we know we could technically add them together. Um, but this is equal to a squared plus eight times B plus B times a plus be squared. So, yes, it is equal to that. And this is with those A B is not equal to be a, um, which is true, because in this case, uh, we don't know what A and R p r. But this is possible because of the fact that matrix multiplication is not communicative. Um, And so now we have our second part, and we have is a plus B squared, always equal Teoh a squared plus Teoh a B plus B squared. And the answer is no. So I'll say no. Um because that would only be true. Um Onley True, when a B is equal to be a um so when a be is equal to be a then a plus, B squared is equal to a plus p times a plus B, and that's equal to, um, a squared plus a B plus be, um plus B squared. And then that would be equal. I want to say one here, um, a skirt plus to a be plus B squared. But this isn't always true, of course, because, as we said earlier, Major specification is non community of And so because of that, um, we could have our A and R B R A B and R B a. Excuse me not equal to each other. So this would only be true if they were both equal to each other.


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