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## More determinant depth

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# Upper triangular determinant

## Video transcript

Let's say I have a matrix where
everything below the main diagonal is a 0. And I'll start-- just for the
sake of argument, let's start with a 2 by 2 matrix. I have the values
a, b, 0, and d. Instead of a c, I have a 0
there, so everything below the main diagonal is a 0. What is the determinant
of this going to be? Let's call that matrix a. The determinant of a is going
to be equal to ad minus b times 0. That's just a 0, so you don't
have to write it. It's equal to a times d. Now, let's say I have
another matrix. Let's call it b. We'll say it's a
3 by 3 matrix. And let's say its entries
are a, b, c. We've got a 0 here. Then you-- let's say you have
a d here, e, then you have another 0 here, another 0
here, and you have an f. Once again, all of the entries
below the main diagonal are 0. What's this guy's determinant? Well, we learned several videos
ago that you can always pick the row in the column that
has the most 0's on it. That simplifies your
situation. Let's find the determinant along
this column right here. The determinant of b is going
to be equal to a times the submatrix if you were to ignore
a's row and column. a times the determinant of d,
e, 0, f, and then minus 0 times its submatrix. You could cancel out-- or times
the determinant of its submatrix, that row
and that column. You'd get b, c, 0, f. And then you have plus 0 times--
you get rid of that row, that column, you
get b, c, d, e. Obviously, these two guys
are going to be 0. I don't care what these 2 by
2 matrices-- what their determinants end up
evaluating to. These are both going to be
equal to 0, because we're multiplying by 0. We're left with a times the
determinant of this, and the determinant of this is pretty
straightforward. We're going to have-- it's just
going to be equal to a times the determinant of this,
which is df minus 0 times e. It's just going to be df. The determinant of b is adf. Notice that the determinant
of a was just a and d. Now, you might see a pattern. In both cases we had 0's below
the main diagonal, right? This was the main diagonal
right here. And when we took the
determinants of the matrix, the determinant just ended up
being the product of the entries along the
main diagonal. And if you think that that's
a general trend that always applies, you are correct. We can do it in the
general case. Let's do it with our
general case. Let's say we have some matrix,
a, and it is equal to a, 1, 1. And you have a, 2, 2. You're going to have
a 0 right there. And then you just keep going all
the way down to a, n, n. In this row, everything's going
to be a 0, except for that last column. This is all a 0 right here. So everything below the main
diagonal is a 0, just like this one, but we're doing it
in the general n by n case. And everything up here is--
well, it doesn't have to be 0. This is a, 1, 2, all
the way to a, 1, n. This is a, 2, n. Keep going down. So everything at the main
diagonal or above isn't necessarily equal to 0. If you wanted to find the
determinant of a, we could do the same thing we did here. We could go down that first
row right there. The determinant of our matrix,
a, is equal to this guy-- a, 1, 1-- times the determinant
of its submatrix. That's going to be a, 2, 2. It goes all the way to a, 2, n,
and then a, 3, 3, all the way to a, n, n. And then, everything down here
is-- these are all 0's. Once again, we have another
situation where all of the entries below the main
diagonal are 0. What's the determinant of
this guy right here? And what-- you might say
hey, what about the rest of that row? Well, the rest of the row is
just a bunch of 0's, just like we had here. 0 times the determinant of its
submatrix, and then that would be a minus and a plus. 0 times the determinant of its
submatrix, so on and so forth. We just have to pay attention
to this term right there. Now, the same argument
we can do here. To find this determinant, we
can just go down that row. The determinant of this is just
going to be equal to-- let's write out-- let's not
forget our a, 1, 1 out there. The determinant of this is going
to be a, 2, 2 times the determinant of its submatrix. Get rid of its row and its
column, and you're just left with a, 3, 3 all the way
down to a, n, n. Everything up here is non-zero,
so its a, 3n. Then everything below the
diagonal, once again, is just a bunch of 0's. Everything down here
is a bunch of 0's. This is another of
what we call an upper triangular matrix. Let me write that down. This whole class, where you
have 0's below the main diagonal, these are called upper
triangular matrices. Matrices, just like that. Now, we keep doing the process
over and over again. If you just keep following this
pattern over and again, now you're going to have the
determinant of this is a, 3, 3 times its submatrix. And every time, the
submatrix is getting smaller and smaller. You'll eventually get to a, 1,
1 times a, 2, 2 times-- all the way to an minus 2 times
a 2 by 2 matrix over here. This is going to be an minus
1, n minus 1, an. Then this is going to be
a sub n minus 1, n. Then you're going to have
a 0 right here. It's just the bottom right-hand
corner of our original matrix, is what you're
going to be left with. And what is the determinant
of this? Well, it's just the product
of these two things. It's just this guy times this
guy minus this guy times that guy, but that's just 0. The determinant of a ends up
becoming a, 1, 1 times a, 2, 2, all the way to a, n, n, or
the product of all of the entries of the main diagonal. Which is a super important take
away, because it really simplifies finding the
determinants of what would otherwise be really hard
matrices to find the determinants of. You could imagine if this
was a 100 by 100 matrix. Now, we could just multiply
the diagonal. Just to make sure that
things are clear, let me do an example. Let's say we find the
determinant of 7, 3, 4, 2. So we have 0's here. This is a minus 2, 1, and a 3,
a 0 here-- sorry, we don't want 0's there. We don't need to
have 0's there. 6, 7-- we actually could have
0's there, but we don't need to have 0's there. And a 0 there, and we
have 0's there. Just like that. So its upper triangular matrix--
if you want to evaluate this determinant, you just multiply
these entries right here. The determinant is equal to 7
times minus 2 times 1 times 3. So it's 7 times minus 6 which
is equal to minus 42. And it's that easy.