Let's do some more matrix multiplication examples,
because I think it is all about seeing as many examples
as possible.
So let's do what may seem to be a more difficult problem,
and it might not be even clear that we can
multiply these matrices.
And maybe that's the first thing we should think about.
So let's say I wanted to multiply the matrix-- I'll do
it relatively small so we don't run out of space.
3, 1, 2, minus 2, 0, 5.
Let's say I want to multiply that times the matrix, minus
1, 0, 2, 3, 5, 5.
I'm just making up these numbers.
So the first thing you might be wondering is, well can I
even multiply these matrices?
Because you know from the first video we did on
matrices, that you can't add these two matrices.
This term corresponds to this one; this one corresponds to
this one; but this term corresponds to nothing over
here so you couldn't add or subtract these matrices.
So the question is, can I multiply these matrices?
Well, what did we learn about multiplying matrices?
We know that, for example, if this is going to result in
some matrix--
However we don't know even what the dimensions are yet
until we work through this example, although there is a
quick way for figuring it out.
So this first term here, the upper left term, where does it
get its row information from and where does it get its
column information from?
Well, it gets its row information from here.
So it is essentially this row times which column?
Times this column, right?
And we can actually take the dot product of this row vector
and this column vector because they have the same length.
This is a column vector but it has a length of 3, right?
But it's a 3 by 1, it has three elements in it.
And this is a 1 by 3 row vector, but it also has three
elements in it.
So we actually can take the dot product or we can
multiply these two.
And similarly we can multiply this times this whole thing to
get this term right here.
And we can multiply this thing times this thing to get this
term, and then this thing times that
term to get that term.
So it actually turns out that you can-- so what kind of a
matrix is this?
Let's call it that this is matrix-- let me switch to
[UNINTELLIGIBLE].
So this is matrix A.
And what are it's dimensions?
It has 2 rows, 1, 2, and 3 columns.
So it's a 2 by 3 matrix.
We're multiplying it times B.
And what are B's dimensions?
Well, it has 3 rows, 1, 2, 3.
So it is a 3 by-- and how many columns does it have?
1, 2-- 2 matrix.
So it turns out that we can multiply two matrices.
You can say that the number-- if, on the first matrix-- the
number of columns is equal to the number of rows in the
second matrix.
So here, 2 by 3 times 3 by 2, we can multiply.
For example, we could have multiplied, if
this is matrix C.
I don't know if I take so much time to keep
bolding these things.
And I don't care how many rows it has.
It can have n rows, n times a columns.
I can multiply it times matrix D, as long as
matrix D has a rows.
As long as you can say these two inner numbers are the
same, right?
This 3 is the same as 3.
And why does that matter, what was the logic?
Because this row will have 3 elements because there's 3
columns, and each column vector here will have 3
elements, because there's 3 rows.
That's where intuition comes from, but if you had to do it
really quickly you say 2 by 3, 3 by 2, this number is equal
to that number, I can multiply.
So let me clear up some space and let's do the
multiplication.
Let's do some multiplication.
I'm debating where I should do it, actually I think I should
do it down here maybe because I'll have more space.
So let me do it down there; I don't have to
erase anything else.
So let me get some space ready.
OK, this will take up a lot of space.
So to get this row 1, column 1 element, what do I do?
I multiply this vector times this vector.
I take the dot product, right?
So it's 3 times negative 1-- I'm just going to write it all
-- 3 times minus 1, plus 1 times 0, plus 2 times 2.
There, we got the first term.
So the second term here, what am I going to do?
I'm going to multiply that vector times that row vector
times this column vector.
And I think you're getting the hang of this, and really the
hardest part about this is staying focused and not making
a careless mistake.
And not getting it confused with rows and
columns and all that.
It just sends blood to your brain, but it's not
that hard, I think.
So what do we do?
We multiply this row vector times this column vector to
get row 1, column 2, right?
Because this row, row 1, column 2.
3 times 3, plus 1 times 5, plus 2 times 5, right.
We're just multiplying the corresponding terms, the third
term times the third term, the second term times the second
term, the first term times the first term.
Although, in this case they're going down, in this case
they're going left and right.
Oh, we add them all up.
OK, so now we're in the second row, and we get our row
information from the first vector-- and let me do a red
that I never use because I think it's kind of tacky, this
red right here.
So I'm going to multiply this row vector
times this column vector.
So it's minus 2 times minus 1, plus 0 times 0,
plus 5 times 2.
We're almost done.
Let me see-- I don't like this color at all-- and now we're
going to multiply this row, because we're in this bottom
row, we're in row 2, column 2, to row 2, column 2.
So it's minus 2 times 3, plus 0 times 5, plus 5 times 5.
And then if we simplify, let's see, this is minus 3 plus 0,
plus 4, so this-- if I have my math correct--
simplifies to 1.
9 plus 5 is 14, plus 10 is 24.
This is 1, 24, and then minus 2 times minus 1 is 2, plus 10,
so this is 12.
And then minus 2 times 3 is minus 6, plus 10-- this is 0--
so minus 6 plus 10 is 4.
So that's interesting.
When I multiplied a 2 by 3 vector times a 3 by 2 vector,
what did I get?
I got a 2 by 2 matrix.
A 2 by 3 matrix, a 2 by 3 times 3 by 2 matrix, I got a 2
by 2 matrix.
And where do you see a 2 by 2?
Well, it's like this got multiplied with this, and what
we have left over is a 2 by 2 matrix.
So in general-- well actually, before I go into the general,
let me ask you a question.
Could I have multiplied the matrices the other way?
Could I have multiplied-- so this right here that is A
times B, or you can sometimes write this AB, and we'd bold
it all up so we know it's matrices.
So could we have multiplied B times A?
Let me clear this down here and let's try.
Let's see if we can multiply B times A.
I think you can already suspect that, since I'm asking
the question, maybe you cannot.
Let's clear up some space.
Let's try to do it the other way around, let's try to
multiply B times A.
So B is minus 1, 0, 2, 3, 5, 5.
And A is-- I'm switching the order-- 3, 1,
2, minus 2, 0, 5.
And I tend to put brackets around my matrices.
Some people have these big parentheses.
It's just all notation; there's nothing particular
about notation.
So let's see if you can multiply these.
So we learned that you get the row information from the first
matrix and the column information
from the second one.
So this term, in theory, should be that row times what?
Well, actually, it turns out that you can multiply them.
Why?
Because this is a 3 by 2, and this is a 2 by 3, right?
So we're going to take that row times what?-- times this
column to get the first term, right?
So what is it going to be?
It's going to be minus 1.
So I actually thought I was doing a counter example, but
actually because this too is the same as this, or when you
switch the row this is the same as this, you
can multiply them.
So I wanted to do a counter example, but hey.
Let's just work through this because it never hurts to see
another example.
And you can see that I just do this on the fly.
So let's do this.
And actually ahead of time, how large will this matrix be?
Well, this is interesting.
It's actually going to be a 3 by 3 matrix,
a much bigger matrix.
Let's work it all out, and maybe you want to pause it and
try it yourself.
This row times this column, so minus 1 times 3, it's minus 3,
3 times minus 2 is minus 6.
And then it's going to be this row times this column.
So it's minus 1 times 1 plus 3 times 0, so that's
just minus 1, right?
Because 3 times 0 is 0.
And then, that was that one, then there's the middle one,
and now we get the row 1, column 3.
So row 1, column 3.
So it's that row times this column.
You can tell, this is often better done by a computer.
Minus 1 times 2 is minus 2 plus 15-- 3 times 5-- so minus
2 plus 5 is 13.
Let's keep going.
So now we're going to take--
I'm sweating, this is so computationally intensive--
We're taking this row times each of these columns.
And actually we are going to learn later that there are
multiple ways of actually thinking about how this
multiplication happens, even multiple ways by computer, but
this is the traditional way.
So this row times each of these columns, right?
So 0, 5, so 0 times 3 plus 5 times minus 2, that's minus
10, and it's 0 times 1 plus 5 times 0.
That's easy, that's 0.
0 times 2 plus 5 times 5 is 25-- almost
there, almost done.
Now we're going to take this row and multiply it times each
of these columns.
So 2 times 3, that's 6, plus minus 10, so
that's minus 4, right?
2 times 3 plus 5 times minus 2.
Yes, that's minus 4, 6 minus 10, right?
We have 2 times 1 plus 5 times 0, that's 2.
Then you have 2 times 2 plus 5 times 5, so 4 plus 25 is 29.
And of course that first term, minus 3 minus 6, so
this is minus 9.
So there you have it.
We multiplied the 3 by 2 matrix times a 2 by 3 matrix,
and we got a 3 by 3 matrix.
And where did that 3 by 3 come from?
Because this 3 is the number of rows in the first matrix,
and this 3 is the number of columns in the second matrix,
which makes sense because we got our row information from
the first matrix and our column information from the
second matrix.
Now let me actually show you an example
that you cannot multiply.
So what if I wanted to multiply a-- let me do a very
simple example-- what if I wanted to multiply
the matrix, 2, 1.
And really, all this is, is a row vector.
And let's say that I wanted to multiply this times-- I don't
know, so this is a 2 by 1.
So then let me say I want to multiply this times-- so let
me think of something-- 1, 2, 3, 4, 5, 6.
Now, can I multiply this?
Well, what do we have?
This is a 3 by 2 matrix.
Can I multiply these two matrices?
Well, what do we have to do?
We get our row information from here, and our column
information from here--
Oh sorry, this isn't 2 by 1, this is 1 row, two columns.
This is a 1 by 2, right?
That's a 1 by 2 matrix.
So can you multiply the 1 by 2 times a 3 by 2 matrix?
So we get our row information from here, so we essentially
have to multiply this by this times this column to get our
first element, then this times this column to
get our second element.
And I don't know what happens from there, but let me-- well,
can we multiply?
Just the way we have defined our multiplication, or the dot
product, can we multiply?
Let's see, 2 times 1 plus 1 times 2.
Then we don't have anything to do with the 3.
So the way that we've defined matrix multiplication, you
cannot multiply these two matrices.
And you didn't have to go through that exercise.
You could've looked at the dimensions, 1 by 2, and this
is a 3 by 2.
This 2 is not equal to this 3, the number of columns in the
first are not equal to the number of rows in the second.
So you can not multiply those two matrices.
So that's something interesting to think about.
And they're actually examples, and it's a good exercise for
you to think about it, where you can multiply A times B,
but you can't multiply B times A.
So I want you think about examples where that happens.
But anyway, I'm pushing 15 minutes, and I will see you in
the next video.