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Studying for a test? Prepare with these 7 lessons on Matrix transformations.
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Say I have some matrix a -- let's say a is n by n, so it looks something like this. You've seen this before, a 1 1, a 1 2, all the way to a 1 n. When you go down the rows you get a 2 1, that goes all the way to a 2 n. And let's say that there's some row here, let's say row i, it looks like a i 1, all the way to a i n. And then you have some other row here, a j, it's a j 1 all the way to a j n. And then you keep going all the way down to a n 1, a n 2, all the way to a n n. This is just an n by n matrix, and you can see that I took a little trouble to write out my row a, my i'th row here and my j'th row here. And just to kind of keep things a little simple, let me just define -- just for notational purposes, you can view these as row vectors if you like, but I haven't formally defined row vectors so I won't necessarily go there. But let's just define the term r i, we'll call that row i, to be equal to a i 1, a i 2, all the way to a i n. You can write it as a vector if you like, like a row vector. We haven't really defined operations on row vectors that well yet, but I think you get the idea. We can then replace this guy with r 1, this guy with r 2, all the way down. Let me do that, and I'll do that in the next couple of videos because it'll simplify things, and I think make things a little bit easier to understand. So I can rewrite this matrix, this n by n matrix a, I can re-write it as just r i. Actually, this just looks like a vector, it's just a row vector. Let me write it as a vector like that. And I'm being a little bit hand-wavy here because all of our vectors have been defined as column vectors, but I think you get the idea. So let's call that r 1, and then we have r 2 is the next row, all the way down. You keep going down, you get to r i -- that's this row right there -- r i. You keep going down, you get r j, and then you keep going down until you get to the n'th row. And each of these guys are going to have n terms because you have n columns. So that's another way of writing this same n by n matrix. Now what I'm going to do here is, I'm going to create a new matrix-- let's call that swapping the swap matrix of i and j. So I'm going to swap i and j, those two rows. So what's the matrix going to look like? Everything else is going to be equal. You have row 1-- assuming that 1 wasn't one of the i or j's, it could have been. Row 2, all the way down to-- now instead of a row i there you have a row j there, and you go down and instead of a row j you have a row i there. And you go down and then you get r n. So what did we do? We just swapped these two guys. That's what the swap matrix is. Now I think it was in the last video or a couple of videos ago, we learned that if you just swap two rows of any n by n matrix, the determinant of the resulting matrix will be the negative of the original determinant. So we get the determinant of s, the swap of the i'th and the j rows is going to be equal to the minus of the determinant of a. Now, let me ask you an interesting question. What happens if those two rows were actually the same? What if r i was equal to r j? If we go back to all of these guys, if that row is equal to this row? That means that this guy is equal to that guy, that the second column-- the second column for that row all the way to the n'th guy is equal to the n'th guy. That's what I mean when I say what happens if those two rows are equal to each other. Well, if those two rows are equal to each other, than this matrix is no different than this matrix here, even though we swapped them. If you swap two identical things, you're just going to be left with the same thing again. So if-- let me write this down-- if row i is equal to row j, then this guy, then s, the swapped matrix, is equal to a. They'll be identical. You're swapping two rows that are the same thing. So that implies a determinant of the swapped matrix is equal to the determinant of a. But we just said, if the swap matrix, when you swap two rows, it equals a negative of the determinant of a. So this tells us it also has to equal the negative of the determinant of a. So what does that tell us? That tells us if a has two rows that are equal to each other, if we swap them, we should get the negative of the determinant, but if two rows are equal we're going to get the same matrix again. So if a has two rows that are equal-- so if row i is equal to row j-- then the determinant of a has to be equal to the negative of the determinant of a. We know that because the determinant of a, or a is the same thing as the swapped version of a, and the swapped version of a has to have the negative determinant of a. So these two things have to be equal. Now what number is equal to a negative version of itself? If I just told you x is equal to negative x, what number does x have to be equal to? There's only one value that it could possibly be equal to. x would have to be equal to 0. So the takeaway here is, let's say if you have duplicate rows-- you can extend this if you have three or four rows that are the same-- leads you to the fact that the determinant of your matrix is 0. And that really shouldn't be a surprise. Because if you have duplicate rows, remember what we learned a long time ago. We learned that a matrix is an invertible if and only if the reduced row echelon form is the identity matrix. We learned that. But if you have two duplicate rows-- let's say these two guys are equal to each other-- you could perform a row operation where you replace this guy with this guy minus that guy, and you'll just get a row of 0's. And if you get a row of 0's, you're never going to be able get the identity matrix. So we know that duplicate rows could never get reduced row echelon form to be the identity. Or, duplicate rows are not invertible. And we also learned that something is not invertible if and only if its determinant is equal to 0. So we now got to the same result two different ways. One, we just used some of what we learned. When you swap rows, it should become the negative, but if you swap the same row, you shouldn't change the matrix. So the determinant of the matrix has to be the same as itself. So if you have duplicate rows, the determinant is 0. Which isn't something that we had to use using this little swapping technique, we could have gone back to our requirements for invertability-- I think was five or six videos ago. But I just wanted to point that out. If you see duplicate rows. and actually if you see duplicate columns-- I'll leave that for you to think about-- if you see duplicate rows or duplicate columns, or even if you just see that some rows are linear combinations of other rows-- and I'm not showing that to you right here-- then you know that your determinant is going to be equal to 0.