Well, after the last video, hopefully, we're a little
familiar with how you add matrices.
So now let's learn how to multiply matrices.
And keep in mind, these are human-created definitions for
matrix multiplication.
We could have come up with completely different ways to
multiply it.
But I encourage you to learn this way because it'll help
you in math class.
And we will see later that there's actually a lot of
applications that come out of this type of matrix
multiplication.
So let me think of two matrices.
I will do two 2 by 2 matrices, and let's multiply them.
Let's say-- let me pick some random numbers: 2,
minus 3, 7, and 5.
And I'm going to multiply that matrix, or that table of
numbers, times 10, minus 8-- let me pick a good number
here-- 12, and then minus 2.
So now there might be a strong temptation-- and you know in
some ways it's not even an illegitimate temptation-- to
do the same thing with multiplication that we did
with addition, to just multiply the corresponding
terms. So you might be tempted to say, well, the first term
right here, the 1, 1 term, or in the first row and first
column, is going to be 2 times 10.
And this term is going to be minus 3 times
minus 8 and so forth.
And that's how we added matrices so maybe it's a
natural extension to multiply matrices the same way.
And that is legitimate.
One could define it that way, but that's not the way it is
in the real world.
And the way in the real world,
unfortunately, is more complex.
But if you look at a bunch of examples I
think you'll get it.
And you'll learn that it's actually fairly
straightforward.
So how do we do it?
So this first term that's in the first row and its first
column, is equal to essentially this first row's
vector-- no, this first row vector--
times this column vector.
Now what do I mean by that, right?
So it's getting it's row information from the first
matrix's row, and it's getting it's column information from
the second matrix's column.
So how do I do that?
If you're familiar with dot product, it's essentially the
dot product of these two matrices.
Or without saying it that fancy, it's just this: it's 2
times 10, so 2-- I'm going to write small-- times 10, plus
minus 3 times 12.
I'm going to run out of space.
And so what's this second term over here?
Well, we're still on the first row of the product vector but
now we're on the second column.
We get our column information from here.
So let's pick a good color-- this is a slightly different
shade of purple.
So now this is going to be-- I'll do that in another
color-- 2 times minus 8-- let me just write out the number--
2 times minus 8 is minus 16, plus minus 3 times minus 2--
what's minus 3 times minus 2?
That is plus 6, right?
So that's in row 1 column 2.
It's minus 16 plus 6.
And then let's come down here.
So now we're in the second row.
So now we're going to use-- we're getting our row
information from the first matrix-- I know this is
confusing and I feel bad for you right now, but we're going
to a bunch of examples and I think it'll make sense.
So this term-- the bottom left term-- is going to be this row
times this column.
So it's going to be 7 times 10, so 70, plus 7 times 10
plus 5 times 12, plus 60.
And then the bottom right term is going to be 7 times minus
8, which is minus 56 plus 5 times minus 2.
So that's minus 10.
So the final product is going to be 2 times 10 is 20, minus
36, so that's minus 16 plus 6, that's 10.
90-- was that what I said?
No, it was-- 70, plus 60, that's 130.
And then minus 56 minus 10, so minus 66.
So there you have it.
We just multiplied this matrix times this matrix.
Let me do another example.
And I think I'll actually squeeze it on this side so
that we can write this side out a little bit more neatly.
So let's take the matrix and now 1, 2, 3, 4, times the
matrix 5, 6, 7, 8.
Now we have much more space to work with so this should come
out neater.
OK, but I'm going to do the same thing, so to get this
term right here-- the top left term-- we're going to take--
or the one that has row 1 column 1-- we're going to take
the row 1 information from here, and the column 1
information from here.
So you can view it as this row vector
times this column vector.
So it results, 1 times 5 plus 2 times 7.
Right?
There you go.
And so this term, it'll be this row vector times this
column vector-- let me do that in a different color-- will be
1 times 6 plus 2 times 8.
Let me write that down.
So it's 1 times 6 plus 2 times 8.
Now we go down to the second row.
And we get our row information from the first vector-- let me
circle it with this color-- and it is 3 times 5
plus 4 times 7.
And then we are in the bottom right, so we're in the bottom
row and second column.
So we get our row information from here and our column
information from here.
So it's 3 times 6 plus 4 times 8.
And if we simplify, this is 5 plus--
Well actually, let me just remind you where all the
numbers came from.
So we have that green color, right?
This 1 and this 2, that's this 1 and this 2,
this 1 and this 2.
Right?
And notice, these were in the first row and they're in the
first row here.
And this 5 and this 7?
Well, that's this 5 and this 7, and this 5 and this 7.
So, interesting.
This was in column 1 of the second matrix and this is in
column 1 in the product matrix.
And similarly, the 6 and the 8.
That's this 6, this 8, and then it's used here, this 6
and this 8.
And then finally this 3 and the 4 in the brown, so that's
this 3, this 4, and this 3 and this 4.
And we could of course simplify all of it.
This was 1 times 5 plus 2 times 7, so that's 5 plus 14,
so this is 19.
This is 1 times 6 plus 2 times 8, so it's 6 plus
16, so that's 22.
This is 3 times 5 plus 4 times 7.
So 15 plus 28, 38, 43-- if my math is correct-- and then we
have 3 times 6 plus 4 times 8.
So that's 18 plus 32, that's 50.
So now let me ask you-- just so you know that the product
matrix-- just write it neatly-- is
19, 22, 43, and 50.
So now let me ask you a question.
When we did matrix addition we learned that if I had two
matrices-- it didn't matter what order we added them in.
So if we said, A plus B-- and these are matrices; that's why
I'm making them all bold-- we said this is the same thing as
B plus A, based on how we define matrix
addition, B plus A.
So now let me ask you a question.
Is multiplying two matrices, is AB-- that's just means
we're multiplying A and B-- is that the same thing as BA?
Does it matter?
Does the order of the matrix multiplication matter?
And so, I'll tell you right now, it actually matters a
tremendous amount.
And actually there are certain matrices that you can add in
one direction that you can't add in the other-- oh, that
you can multiply in one way but you can't multiply in the
other order.
And well, I'll show you that in an example-- but just to
show that this isn't even equal for most matrices, I
encourage you to multiply these two matrices in the
other order.
Actually let me do that.
Let me do that really fast just to prove
the point to you.
So let me delete all this top part.
Let me delete all of it, and actually I can delete to this.
So hopefully, you know that when I multiply this matrix
times this matrix, I got this.
So let me switch the order-- and I'll do it fairly fast
just so as to not bore you-- so let me switch the order of
the matrix multiplication.
This is good as this is another example-- so I'm going
to multiply this matrix: 5, 6, 7, 8, times this matrix-- and
I just switched the order; and we're testing to see whether
order matters-- 1, 2, 3, 4.
Let's do it-- and I won't do all the colors and everything,
I'll just do it systematically.
I think you just have to see a lot of examples here-- So this
first term gets its row information from the first
matrix, column information from the second matrix.
So it's 5 times 1 plus 6 times 3, so it's 5 times 1--
Let me just write, actually edit.
I'm going to skip a step here-- OK so it's 5 times 1
plus 6 times 3, plus 18.
What's the second term here?
It's going to be 5 times 2 plus 6 times 4.
So 5 times 2 is 10, plus 6 times 4 is 24.
Right, now we just took this row times this
column right here.
OK now we're down here for the set-- so then we're doing this
row, this element right here at the bottom left is going to
use this row, and this column.
So it's 7 times 1 plus 8 times 3.
8 times 3 is 24.
And then finally, to get this element we're essentially
multiplying this row times this column, so it's 7 times 2
is 14, plus 8 times 4, plus 32.
So this is equal to 5 plus 18 is 23, 34.
What's 7 plus 24?
That's 31, 46.
So notice, if we called this matrix A and this
is matrix B, right?
In the last example, we showed that A times B is equal to 19,
22, 43, 50.
And we just showed that, well, if you reverse the order, B
times A is actually this completely different matrix.
So the order in which you multiply
matrices completely matters.
So I'm actually running out of time.
In the next video I going talk a little bit more about the
types of matrix-- well, one, we know that order matters--
and in the next video I'll show that what type of
matrices can be multiplied by each other.
When we added or subtracted matrices, we just said, well
they have to have the same dimensions because you're
adding or subtracting corresponding terms. But
you'll see with multiplication it's a little bit different.
And we'll do that in the next video.
See you soon.