Transcript:

Alright, thanks for watching and today. I want to proof another. Very useful identity about determinants, namely that the determinant of a transpose equals to the determinant of a and then at the end, I will also solve a little mystery of linear algebra, so just think about all right, so we want to show this. How do we do this? First of all, just actually like the proof. The determinant of a B is determinant of a times the determinant of B. Let’s get rid of the case where a is not invertible so note. If this one is not invertible, well, then only one hand. We know that the determinant of a is zero and we can independently show this, but also if a is not invertible, then the rank of a is not optimal, so it’s strictly less than M for AIDS N by N, But then what is the rank of a transpose? Well, I definitely not by definition, but it is true that it’s the same as the rank of a and if you want it to seem as a number of pivots, OK, so it turns out the number of pivots in a is the same as the number of pivots in a transpose. And that’s if you want because a pivot row of a is a pivot column of a transpose pivots only to get changed, so the rank of a transpose is the rank of a and, in particular because the rank of a is less than N. It follows that the rank of a transpose is less than N. And, in particular, this implies that a transpose is not invertible and particularly we get that the determinant of a transpose. It’s also 0 which is so in the case where a is not invertible. We know that this identity holds, and therefore, from now on assume that a is invertible and just like before for the second case, let’s do the simplest case possible, so that’s assume a is an elementary matrix. I want to show you that. In each case, we do have the terminal. A equals the term of a transpose so first of all, assume a is of type one, which just means a interchange’s, two rows interchanges rows by MJ. What it looks like it looks like the identity matrix but grows. I and J are interchanged. So let me just give you an example. If you have a four by four matrix and you interchange rows 1 & 4 then it looks like this 0 0 1 0 1 0 0 0 0 And what do you notice about this matrix? Well, precisely, a transpose equals a in fact, this is always true. The row interchange matrix is always symmetric. And that’s because what does this matrix look like, though it’s the identity, except the IJ entry and the JI entry are 1 and in particular. If you take the transpose, it’s just means the JI entry and the IJ entry will also be 1 which is the same matrix. So in this case because a transpose equals a it follows that trivially determine a transpose equals to the determinant of a so in this case we do have this is true, and by the way that determinant is 1 in that case now time to suppose a multiplies a row by let’s say K. Then what does a look like well? It looks like the identity except row. K has this value K. So the K comma cave entry is K. And in that case, we also have that a transpose equals a. So it’s also true that the determinant of a transpose equals the terminal. A in this case is K. OK now! The trickiest one is time 3 So, hey, if you want abs. K Time’s Rho. I to Rho J so again. Let me give you an example. If the ah let’s see 1 0 0 0 1 0 and you add three times like this. K Times Row 1 to Row 3 It looks like the identity, except you add, you know. K on the 3 comma first entry. But then what does a transpose look like? It still looks like the identity, but except the kids in the wrong position, but that is not a problem because first of all. I have done this in another video that if you add K times one row to the other one that the turbulent is one is one when ever again. You add K times one row to the other, but look, this matrix is of the same form, because in this example, you add three times the first row to the third row. In this example, you add three times the third row to the first row. So a transpose is an elementary matrix of the same type and remember what I said. Whenever you add a multiple one row to the other one. The determinant is still one. So in this case, the determinant of a transpose is also one and therefore it is the same determine of a transpose is the determinant a wonderful, so for all the three times, we have verified that the determinant of a transpose is the determinant of a and now we can move on to the general case. Okay, and again here is where we need. The fact that is invertible, so since a it’s invertible, we know that a is a product of elementary matrices, so yeah, – 1 e 2 E 1 Let’s calculate a transpose. That is the M en minus 1 e 2 E 1 transpose. Now, what does it transpose? Do just like Inverses? It flips everything you want. Transpose E 2 transpose up to E N transpose, which is good because this looks kind of like the wrong order, and now we put it in the right order, and then what is that term of a transpose? It’s a determinant of E 1 transpose, e 2 Transpose N Transpose. It’s almost like ET wants to go home, except with this guys, and here’s a thing. Now we’ve already shown that determine is multiplicative so this is really determine of E 1 transpose determinant of E 2 transpose up to the determine yin transpose, all right, but the whole point is all those guys are elementary. So by case two, we have shown that we can just remove the transpose e1 and the determine kinds of determinate. Yeah, and the cool thing is even though matrices don’t always commute A B is not be a those are all numbers, so it’s not burst. We can just commute them, so this will then become determinant of P N minus 1 determinant of e1 and then again because it’s multiplicative that is the internal up to the determine b1 but that’s precisely the determinant of a so what have we shown? We’ve shown that, indeed, the determinant of a transpose equals blah blah, blah, blah, blah, blah equals to the determinant A so this is the general case, so from now on, feel free, just use this identity, whatever and honestly as promised. I want to solve one of the big mysteries of linear algebra because how did we define the determinant? We defined it as cofactor expansion along the first row and then with a lot of work, we’ve shown that you can actually Cofactor expanded along any row, but it’s not obvious yet that you can expand it out on any column, but this is precisely what we’re gonna show now so so far can evaluate the determinant determining. Hey, using any column, not just any rope or any column. And why is this true we’ll expansion using column? J of a here’s so suppose you have the Jade column and you expand just you expand it. You don’t know if it’s a determinant of not, but suppose you do a cofactor expansion along column J. But look, the Jade column of a is the Jade row of a transpose. So what is this is the expansion using row J of a transpose. But remember whenever you expand along a row, it gives you the determinant. So this expansion calculates indeed the determinant of a transpose, but we’ve just shown that the determinant of a transpose is the determinant of a so working backwards. What happen to have we found? We found that the determinant of a is indeed, the expansion, using any column of a so from now feel free just to expand the main determinant using whichever row of whichever column You want and you can like sleep in peace now, all right. I hope you like this. If you wanna see more math, please make sure to subscribe to my channel. Thank you very much.