Linear Algebra: 3x3 Determinant

In the last video we defined the
notion of a determinant of

a 2 by 2 matrix.

So if I have some matrix-- let's
just call it B-- if my

matrix B looks like this, if
its entries are a, b, c, d,

we've defined to determinant
of B.

Which could also be written as
B with these lines around it,

which could also be written as
the entries of the matrix with

those lines around
it, a, b, c, d.

And I don't want you to
get these confused.

This is the matrix when
you have the brackets.

This is the determinant of the
matrix, when you just have

these straight lines.

And this, by definition, was
equal to ad minus bc.

And you saw in the last video,
or maybe you saw in the last

video, what the motivation
for this came from.

When we figured out the inverse
of B, we determined

that it was equal to 1 over ad
minus bc times another matrix,

which was essentially these
two entry swaps, you

got a d and an a.

And then these two entries
made negative, so

minus c and minus b.

This was the inverse of b.

And we said, well, when
is this defined?

This is defined as long as this
character right here does

not equal 0.

So you said hey, this looks
pretty important.

Let's call this thing right
there the determinant.

And then we could say that B is
invertible, if and only if,

the determinant of B
does not equal 0.

Because if it equals 0, then
this formula for your inverse

won't be well defined.

And we just got this from our
technique of creating an

augmented matrix whatnot.

But the big take away is we
defined this notion of a

determinant it for
a 2 by 2 matrix.

Now the next question is, well
that's just a 2 by 2,

everything we do in linear
algebra, we like to generalize

it to higher numbers of
rows and columns.

So the next step, at least--
let's just do baby steps--

let's start with a 3 by 3.

Let's define what its
determinant is.

So let me construct a
3 by 3 matrix here.

Let's say my matrix A is equal
to-- let me just write its

entries-- first row, first
column, first row, second

column, first row,
third column.

Then you have a2
1, a2 2, a2 3.

Then you have a3 1, third
row first column, a3

2, and then a3 3.

That is a 3 by 3 matrix.

Three rows and three columns.

This is 3 by 3.

I am going to define the
determinant of A.

So this is a definition.

I'm going to define the
determinant of this 3 by 3

matrix A as being equal to--
and this is a little bit

convoluted, but you'll get the
hang of it eventually.

In the next several videos we're
just going to do a ton

of determinants.

So it just becomes a bit of
second nature to you.

It's a little computationally
intensive sometimes.

But it equals this first row.

It equals a1 1 times the
determinant of the matrix you

get, if you get rid of this
guy's column and row.

So if you get rid of this guy's
column and row, you're

left with this matrix here.

So times the determinant of the
matrix a2 2, a2 3, a3 2,

and then a3 3.

Just like that.

So that's our first entry
and that's a plus this.

And then I said it's a plus
this, because the next entry's

going to be a minus.

You have a minus this
guy right here.

So then you're going to have
minus a1 2 times the matrix

you get if you eliminate
his column and his row.

So times, you're going to get
these entries right there.

So a2 1, a2 3, a3 1, and
then you have a3 3.

We're not quite done.

You could probably guess with
the next one's going to be.

Then you're going to have a
plus-- let me switch to a

better color-- plus this guy.

Plus a1 3 times the determinant
of its-- I guess

you could call it--
its sub-matrix.

We'll call it that for now.

So this matrix right here.

So a2 1, a2 2, a3 1, a3 2.

This is our definition of the

determinant of a 3 by 3 matrix.

And the motivation is, because
when you take the determinant

of a 3 by 3 it turns out-- I
haven't shown it to you yet--

that the property is the same.

That if the determinant of this
is 0, you will not be

able to find an inverse.

And when I defined determinant
in this way.

If the determinant does not
equal 0, you will be able to

find an inverse.

So that's where this
came from.

And I haven't shown
you that yet.

And I might not show you
because it's super

computational.

It'll take a long time.

It'll be very hairy and I'll
make careless mistakes.

But the motivation comes from
the exact same place as the 2

by 2 version.

But I think what you probably
want to see right now is at

least just this thing applied
to an actual matrix, because

this looks all abstract
right now.

But if we do it with an actual
matrix, you'll actually see

it's not too bad.

So let's leave the definition up
there, and let's say that I

have the matrix 1, 2, 4, 2, 2,
minus 1, 3, and 4, 0, 1.

So by our definition of a
determinant, the determinant

of this guy right here-- so
let's say I call that matrix

C-- C is equal to that.

So if we want to figure out
the determinant of C, the

determinant of C is equal to--
I take this guy right here,

let me take that 1-- times the
determinant of-- let's just

call it the submatrix,
right here.

So we have a minus 1, we
have a 3, we have a 0,

and we have a 1.

Just like that.

Notice, I got rid of
this guy's column

and this guy's row.

And I was just left with
minus 1, 3, 0, 1.

Next, I take this guy.

And this is the trick.

You have to alternate signs.

If you start with a positive
here, this next one's going to

be a minus.

So you're going to have minus 2
times the submatrix-- we can

call it-- if we get rid
of this guy's column

and this guy's row.

So 2, 3, 4, 1.

I just blanked this out.

If I could videotape my finger,
I would cover my

finger over this column right
here and over that row, and

you'd just see a 2,
a 3, a 4, and a 1.

And that's what I
put right there.

And then finally, we went
plus, minus, plus.

So finally, we'll have plus 4
times the determinant of the

submatrix, if you get rid of
that row in that column.

So 2, minus 1, 4, 0.

Now, these are pretty
straightforward.

These are not too
bad to compute.

Let's actually do it.

So this is going to be equal
to 1 times what?

Minus 1 times 1.

Let me just write it out.

Minus 1 times 1, minus
0 times 3.

This just comes from the
definition of a 2 by 2

determinant.

We've already defined that.

And then we're going to have
a minus 2 times 2 times 1,

minus 4 times 3.

And then finally, we're going to
have a plus 4 times 2 times

0 minus minus 1 times 4.

I wrote it all out
so you can see.

This thing right here is just
this thing right here.

And then you have
the 4 out front.

This thing right here was just
this thing right here.

So it's the determinant of the
2 by 2 submatrix for each of

these guys.

And if we compute this, this is
equal to-- minus 1 times 1

is minus 1.

Minus 0, that's 0.

So this is a minus 1 times
1, so that's a minus 1.

And then we get-- what
is this equal to?

This right here is 12.

So you get 2 minus 12.

Right?

You get 2 times 1
minus 4 times 3.

So it's minus 10.

So that is equal to minus 10.

And then you have a minus
10 times a minus 2.

So that becomes a
plus 20, right?

Minus 2 times minus 10.

And then finally, in the green,
we have 2 times 0,

that's just a 0.

And then you have minus 1 times
4, which is minus 4.

Then you have a minus sign
here, so it's plus 4.

So this all becomes a plus 4.

Plus 4 times 4 is
16, so plus 16.

And what do we get when
we add this up?

We get 20 plus 16 minus 1.

It is equal to 35.

We're done.

We found the determinant
of our 3 by 3 matrix.

Not too bad.

Right there, so that is equal
to the determinant of C.

So the fact that this isn't
0 tells you that C is

invertible.

In the next video, we'll try
to extend this to n by n

square matrices.