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# Determinant after row operations

Video transcript

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. Right? 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.