Determinant after row operations What happens to the determinant when we perform a row operation
Determinant after row operations
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- I have a matrix A.
- It is an n by n matrix.
- And let me just write its rows like this.
- Let me just write it as r1.
- We could call them row vectors maybe. r2, I'm not doing it
- too formally.
- This is just to save on writing.
- And then it has an ith row, ri, and then you
- can keep going .
- That's an i right there.
- Then it has a jth row, rj, and you keep going and you get to
- the nth throw.
- It has n rows and n columns.
- So you get to rn just like that.
- That is my matrix.
- Just to make sure you get what I'm saying, so if I have a kth
- r-sub-k is equal to ak1.
- Maybe I'll write it as a vector.
- ak2 all the way to akn.
- So this is just your standard representation.
- I wrote it this way because we're just going to be dealing
- with rows in this video and it makes our notation a little
- bit easier.
- Let me focus on these two rows right here.
- And let me define another matrix B that is
- also an n by n matrix.
- And it's identical to matrix A except for one row.
- So it's identical to matrix A except for one row.
- You have r1 just like that; it's the
- same as that one there.
- r2, keep going, go down to our ri, even that one's identical.
- But rj I've now replaced.
- I'm replacing rj with rj minus a scalar multiple of ri.
- Minus c times ri.
- So minus a scalar multiple of that.
- I've replaced rj with that.
- So this is equivalent to the row operations we do we did
- our Gaussian Elimination, or when we put things in reduced
- row echelon form.
- And everything else in this matrix is the same as A.
- It's all the way down to rn.
- This is our matrix B.
- So let's think about what the determinant of B is going to
- be equal to.
- I'll do it in blue.
- Well, you could immediately say that B is equivalent to--
- Well, you can imagine two vectors.
- You can imagine two matrices.
- One matrix that look like this.
- One matrix that look like r1, r2, all the way down ri, all
- the way down to rj.
- And then you keep going down to rn.
- That's one matrix, which you may have already noticed is
- identical to A.
- That's one matrix.
- Then you could have another matrix here
- that looks like this.
- It's identical everywhere.
- r1, r2, ri.
- Some dots there to show you I might have skipped some rows.
- Skip some more rows.
- And then you have c times times ri.
- c times ri.
- Let me do that in a different color.
- This is ri right here.
- And then you just keep going down to rn.
- Now, the determinant of B, you could view as the determinant
- of this guy.
- Let me write this here.
- The determinant of B is equal to the determinant of this guy
- plus the determinant of this guy.
- Hopefully, you remember a couple of videos ago, that if
- one matrix-- Let's have two matrices that are identical in
- every way except for one row.
- So these two matrices are completely identical except
- for what's going on on the jth row.
- Here you have a r-sub-j.
- Here you have a c times r-sub-i.
- So it's a scalar multiple of a row that you had
- up here, this guy.
- So this is ri, this is the ith row.
- Here you have an ri, here you have an ri.
- But here you have another version of r row, scalar
- multiple of ri, while here you have an rj.
- Now, if you have another matrix that is essentially
- identical to these two matrices, except
- for this one row.
- And in that one row, it looks like the addition of these two
- matrices-- and let me put a negative here.
- So if you kept this matrix completely identical, but if
- you were to replace it with the sum of these two rows.
- So rj minus c times ri, you'll get this matrix right here.
- You'll get matrix B.
- And we learned that the determinant of B is equal to
- the determinant of this guy and that guy.
- Remember, B is not the sum of these two matrices.
- B is identical to these two matrices, except for that one
- row where B's jth row is equivalent to the jth row of
- this guy, plus the jth row of that guy.
- And when I talk about adding rows, you're just adding their
- corresponding elements.
- So I could rewrite this so this row would look like-- The
- first term would be aj1 minus c times ai1.
- That would be the first term in that row.
- The second term of that row would be aj2
- minus c times ai2.
- And it would go all the way to ajn minus
- ca-sub-in, the nth column.
- So that's all it means by that.
- So the determinant of B is equal to the determinant of
- this plus the determinant of this.
- The determinant of this, this thing right here
- is our matrix A.
- This is going to be the determinant of A.
- And what's the determinant of this?
- Well, let's break this down a little bit more.
- The determine of this is equal to what?
- This is completely equivalent to A, except one of its rows--
- Sorry, this is completely equivalent to this matrix.
- Not equivalent to A.
- Be very careful.
- Don't listen to everything I say.
- It's not equivalent to A.
- The difference is that A has an rj here.
- This guy has a minus c times ri.
- So this is equivalent to this matrix.
- It's completely equivalent to this matrix right here.
- Let me do it like this.
- So you have an r1, r2, keep going, and you have an ri,
- then you have another ri.
- Let me clean this up a little bit.
- Let me clear this out just so I have some
- space to work with.
- You have an ri.
- You have that ri there.
- Then you have another ri.
- You have another ri right there.
- You have another ri.
- So the jth row has an ri there.
- Then you keep going and you have an r-sub-n.
- These two guys are completely equivalent except for this guy
- has a minus c times the jth row.
- That's what this was, right here.
- This is the jth row.
- Everything we're doing is in the jth row.
- This has a minus c times the jth row.
- So the determinant of this guy right here-- Let me just be
- clear that I'm only taking the determinant of
- this guy right here.
- It's going to be equal to minus c times the determinant
- of-- let me write it this way --minus c times the
- determinant of r1, r2.
- You have your first ri.
- And then in the jth row you have another
- version of the ri.
- And then you go down to r-sub-n.
- So times that determinant.
- This is just the determinant of this.
- I've added brackets and straight lines.
- And we saw this a couple of videos ago.
- If you have a matrix, you just multiply one of its rows by a
- scalar, in this case minus c.
- It's equivalent to minus c.
- The determinant of the new matrix is equal to minus c
- times the determinant of your matrix.
- That's all I'm saying right here.
- But what is the determinant of this matrix?
- You might have already noticed that it has duplicate rows.
- It has an ri, and then in the ith row, then it has another
- ri in the jth row.
- Remember, we kind of decomposed this B matrix right
- here as the sum of-- Or its determinant can be described
- as the determinant of the sum of these two things.
- B isn't the sum of these two things.
- Every other element is identical to every other
- element in each of these guys.
- But this guy right here, he has duplicate ri's.
- And what do we know about the determinant of a matrix with
- duplicate entries?
- The determinant is zero.
- So this entry right here is zero.
- Minus c times 0, 0.
- So the determinant of this whole thing is 0.
- So the big take-away right here is that the determinant
- of B is equal to just the determinant of this thing,
- which was the determinant of A.
- This is a very big take-away.
- It's going to make our life very easy.
- The determinant of B is equal to the determinant of A.
- So if you start with some matrix, and you replace the
- jth row in this example, but any row.
- If you replace any row with that row minus some scalar
- multiple of another row-- we picked ri in this case, that
- would be ri --the determinant will not be changed.
- You have to be very particular about how you say it because,
- obviously, if you just multiplied something by a
- scalar-- if you were to change its determinant, or if you do
- other things.
- If you just take a row, if you take the jth row, and you
- replace it with the jth row minus c times the ith row
- times some other row, which is equivalent to just a row
- operation that we have been doing, then it will not change
- your determinant.
- Which is a very big take-away because now we can carefully
- do some row operations and know that the determinant will
- not change.
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